statement by the proposer highlighting major scientific...

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Statement by the proposer highlighting major scientific contributions made by the nominee (NOT TO EXCEED 100 WORDS)

Main work concerns Chern-Simons theory and its relations to topological strings, knot polynomials. One of the challenging problems was to obtain colored HOMFLY polynomials for non-torus knots and links.The nominee’s work resulted in a neat proposal for a simplest class of SU(N) quantum Racah coefficients. Thus leading to the important development of obtaining colored HOMFLY for many non-torus knots and links.

In other interesting research pertaining to existence of at least one quiver

Chern-Simons for every toric Calabi-Yau four-folds, the nominee systematized the procedure for inverse algorithm crucial to determination of the quivers corresponding to complex cones over Fano 3-folds.

Areas of specialisation : (mention three)

Nomination to be considered by the Sectional Committee for : II

(Please choose the most appropriate

subject listed below)

List of Sectional Committees

I. Mathematics IV. Engineering & Technology VII. Animal/Plant Sciences

II. Physics V. Medicine VIII. General Biology

III. Chemistry VI. Earth & Planetary Sciences

1.

Chern-Simons field Theory

2.

Topological String Theory

3.

Quiver gauge theories

2

BIOGRAPHICAL INFORMATION (of the nominee)

Name (expand initials) Pichai Ramadevi

Present position Professor Gender M F

Date of birth 14th May 1968

Address Department of Physics

Indian Institute of Technology Bombay

Mumbai- 400076

Phone: Off. 91-22-25767563

Res. 91-22-25768563

Fax 91-22-

25723480 Email [email protected]

Mobile 9820325207 Personal Homepage

http://www.phy.iitb.ac.in/ doku/doku.php/faculty/ramadevi/home

Academic qualifications

Year Degree University/Institution

1989 BSc Physics Madras University

1991 MSc Physics IIT Madras

1996 PhD Institute of Mathematical Sciences

Positions held (in chronological order)

Year(s) University/Institution

Position held

1996-98 TIFR Mumbai Post-doctoral position

1998-99 HRI Allahabad Post-doctoral position

2000-2001 Physics Department , IIT Bombay

Senior Research Associate(CSIR POOL Scheme)

2001(Mar-May) CALTECH Visiting research associate

2001-present Physics, IIT Bombay Faculty

√

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Awards and Honors

Received IIT Bombay `Best Research Paper Award’ in 2006 for the explicit calculations

which was essential to validate Ooguri-Vafa conjecture.

List of the five most important papers published by the nominee from his/her independent career:

please give: 1) title of the paper

2) names of all authors in the same sequence as it appears in the papers

3) full journal reference

1. Title: U(N) Framed Links, Three-Manifold Invariants, and Topological Strings

Authors: Pravina Borhade, P. Ramadevi, Tapobrata Sarkar

Journal Ref: Nuclear Physics B 678, (2004), 656-681.

2. Title : SO(N) Reformulated Link Invariants from Topological Strings

Authors: Pravina Borhade, P. Ramadevi

Journal Ref: Nuclear Physics B727 (2005) 471-498.

3.

Title: SU(N) quantum Racah coefficients & non-torus links

Authors: Zodinmawia and P. Ramadevi,

Journal Ref: Nuclear Physics B870 (2013) 205-242.

4. Title: Inverse algorithm and M2-brane theories

Authors: Siddharth Dwivedi and P. Ramadevi

Journal Ref: Journal of High-Energy Physics 1111 (2011) 111

5.

Title: Multiplicity-free quantum 6j-symbols for Uq(slN)

Authors: Satoshi Nawata, P. Ramadevi, Zodinmawia

Journal Ref:Letters in Mathematical Physics. 103 (2013) 1389-1398.

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Two-Page summary of the Scientific Contributions

Chern-Simons field theoretic invariants:

The nominee attempted one of the challenging problems, called the `Classification

problem of knots’, within the Chern-Simons field theoretic framework as a part of her PhD thesis. Interestingly, the work with her collaborators on detection of chirality of knots showed that the Chern-Simons field theory invariants of knots are definitely more powerful than the well-known Jones, HOMFLY and Kauffman polynomials. Further, they gave an analytic proof that the field theoretic invariants cannot distinguish a class of knots called mutant knots. Recently, the nominee with her student exploited knot equivalences to determine some Racah coefficients to explicitly write the polynomial form for invariants of many non-torus knots and links(paper 3). Three-Manifold Invariants:

A process known as surgery of knots inside the three-sphere gives rise to new three-manifolds. The nominee was also involved in obtaining an algebraic expression for three-manifold invariants in terms of Chern-Simons invariants of the corresponding knots inside three-sphere S3. These three-manifold invariants were shown to be proportional to the Chern-Simon partition function for Lens spaces L(p,q) and Poincare manifold.

Topological strings Vs Chern-Simons theory

Gopakumar-Vafa duality conjecture states that the Chern-Simons theory on S3 is dual to A-model topological string theory on a resolved conifold. The conjecture was verified by showing that the free-energy of Chern-Simons partition function on S3 matched with the string partition function. Subsequently, Ooguri-Vafa verified the conjecture for the simplest observable (unknot) leading to a conjectured form for other observables which is known as `Ooguri-Vafa conjecture.’ The nominee with her collaborator verified the conjectured form for many knots upto 7 crossings which received a lot of attention internationally. To extend Gopakumar-Vafa conjecture to other three-manifolds, the nominee with her student attempted a meaning for surgery in the topological string context. Interestingly, the large N expansion(`t Hooft proposal) of the Chern-Simons free energy for some manifolds resembled topological string partition function whose target space is Calabi-Yau with one Kahler parameter(paper 1). Then, she explored, with her student, the presence of orientifolds in topological strings from SO(N) Chern-Simons theory(paper 2). The oriented contribution to such topological strings in orientifold backgrounds required the invariants of knots carrying composite representations. The nominee with her collaborators obtained the composite invariants from Chern-Simons approach for torus knots. Homological Invariants

The Laurent series of the SU(N) Chern-Simons field theoretic knot invariants are always polynomials in two variables which are functions of rank N and coupling constant k. Surprisingly, the coefficients in the Laurent series are integers. These integers must be given a topological meaning. Khovanov introduced a categorification approach by

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presenting a bi-graded homological chain complex Hi,j . He showed that the integers are the dimensions of Hi,j. The nominee with her collaborators has conjectured slN homological invariants for a class of knots called twist-knots. This led to the study of generalized volume conjectures and obtaining quantum A-polynomials. They have also studied SO(N) homological invariants for some knots in a recent paper. Quantum slN Racah coefficients

From the results on knot polynomials, there appeared to an indirect way of determining quantum slN Racah coefficients. The nominee with her collaborators succeeded in obtaining a closed form expression for Racah coefficients which involves simplest class of representations(paper 5). This result is definitely a breakthrough result. Quiver Chern-Simons theory Vs toric data of Calabi-Yau 4-folds (CY4)

This research area addresses three dimensional quiver Chern-Simons gauge theories on coincident M2 branes at the tip of singular CY4 dual to M-theory on AdS4 * X7 , where real cone over X7 (known as Sasaki-Einstein manifolds) gives CY4. The nominee with her student reviewed the literature on inverse and forward algorithm of finding quiver gauge theories from the toric data of CY4 and the converse. There were 18 toric Fano 3-folds whose complex cones gives CY4. They systematized the inverse algorithm to find quiver gauge theories for four Fanos: B1, B2, B3, P3(paper 4). The results suggests that there will be at least one quiver for every toric CY4. Later on, they studied partial resolution to check the embeddings inside CY4 which are complex cones over Fano B. Recently, they have computed the superconformal index for these theories to show the two quivers corresponding to Fano B2 are not Seiberg-like dual theories. Non-Supersymmetric States in String Theory

During her post-doctoral tenure, she collaborated with various people and wrote interesting papers. One of them was to show mass non-renormalisation of a non-supersymmetric state which justifies the entropy matching with microstate counting. An indirect method involving boosting Schwarzchild string in one-dimension higher leads to charged black holes in one-lower dimensions was exploited to do the counting of microstates of non-supsersymmetric Schwarzchild black hole. They further did an elaborate study of Hawking radiation of four-dimensional Schwarzchild black holes. She also showed her independence by writing a single author paper where she tried to find supergravity solution for three-string junction in M-theory. Manpower training, Popularizing Science

The nominee has also written three popular articles in Resonance journal and one article in Physics News magazine. In 2005, she organized a one day celebration of world year of Physics where the partipants were from various colleges in Mumbai. She organized SERC school in 2008 and has taken tutorials in some of these schools leading to training many young graduate students.

CURRICULUM VITAE

Name: Pichai Ramadevi

(family name) (first name)

Sex: Female

Date of Birth: May 14, 1968

Place of Birth: Chennai, India

Nationality: Indian

Marital Status: Married

Present Address: Physics Department

Indian Institute of Technology, Bombay

Mumbai 400 076, India

E-mail address: [email protected]

Telephone: 91-(0)22-25767563

PRESENT POSITION

Professor, Department of Physics, IIT Bombay from June 2012 onwards.

ACADEMIC RECORD:

• Master of Science in Physics from I.I.T. Madras (July 1989-June 1991), CGPA - 9.06/10.0.

• Ph.D. in Physics from the Institute of Mathematical Sciences, Chennai (August 1991-

August 1996)

Thesis Title : Chern-Simons theory as a theory of Knots and Links

Thesis Advisor: Prof. T.R. Govindarajan.

• Visiting Fellow (Post-Doctoral Fellow) at Tata Institute of Fundamental Research, Mum-

bai (September 1996- August 1998).

• Visiting Fellow (Post-Doctoral Fellow) at Mehta Research Institute, Allahabad (Septem-

ber 1998- Jan 1999).

• Visitor at Tata Institute of Fundamental Research, Mumbai (July 1999- Oct 1999).

• Senior Research Associate (CSIR), Physics Department, Indian Institute of Technology,

Bombay (Jan 2000-Feb 2001).

• Visiting Associate at CALTECH, USA (Mar - May 2001).

• Assistant Professor, Department of Physics, IIT Bombay (June 2001 - April 2007)

• Associate Professor, Department of Physics, IIT Bombay (April 2007- June 2012)

FIELDS OF INTEREST:

• Chern-Simons Field Theories, Knot theory and connections to Topological String Theo-

ries.

• String Theory and its application to Black Hole Physics, Non-Supersymmetric States in

String Theory.

• Matrix models, Supersymmetric gauge theories, quiver gauge theories.

• AdS-hydrodynamics, AdS-CMT

PHD THESIS SUPERVISION:

S.No Name of the Title of Thesis Doctorate Year of

research scholar or Master’s Completion

1 Pravina Borhade Topological Strings Doctorate Sep 2007

& Gauge Theories

2 Brijesh Kumar Topological objects& Doctorate Sep 2011

vacuum stability in QFT

Ongoing PhD Students

S.No Name of the Joining date Research topic

3 Siddharath Dwivedi July 2009 Working on M2branes

4 Zodinmawia Joined July 2010 Working on Chern-Simons Theories& Knots

5 Lata Joshi Joined Jan 2011 Working on AdS-hydrodnamics

TEACHING EXPERIENCE

S. No Title of Course Taught Postgraduate

or

Undergraduate

1 Quantum Mechanics III(advanced) Undergraduate

2 Physics III: Undergraduate

Quantum Physics & Applications

3 Astrophysics Undegraduate

4 Quantum Mech. I Undergraduate

5 Elementary Particle Physics Undergraduate

6 Prep Course Mechanics Undergraduate

7 General Theory of Relativity Undergraduate

8 Elec. & Magnetism: Tut Undergraduate

8 Group Theoretic Methods Undergraduate

9 Mathematical Physics I:Tut Undergraduate

SPONSORED PROJECTS UNDERTAKEN(INDIA):

Sponsoring Title of project Amount of grant Period Co-investigator

Agency (if any)

DST : SERC Chern-Simons Field Theory Two lakhs ten Jan 2005 to None

FAST TRACK &Topological String Theory Thousand Dec 2007

Research Paper Quiver Gauge Theories Five lakhs July 2008 to None

Award Scheme from Dimer model approach Aug 2013

SHORT TERM COURSES/WORKSHOPS/SEMINARS

ETC. ORGANIZED:

1. Member of the National organising committee for the International Workshop WHEPP

held at IIT Bombay during Jan 2004.

2. I organised a one day celebration of the “WORLD YEAR OF PHYSICS” during Oct 2005

which included interesting popular seminars, laboratory visits, video show and a panel

discussion.

3. I organised DST funded SERC school on High Energy Physics for the early Ph.D students

at IIT Bombay during Feb 2008 (Feb 6-26, 2008).

4. I organised a National String Workshop for research discussions and interaction from

Feb 10th to Feb 15th, 2010.

Invited Lectures in India/Abroad

1. Invited to University de Santiago Compostella for collaboration and to give a seminar

(April 1997).

2. Invited speaker in the Indo-Russian Workshop held at Institute of Mathematical Sciences,

Chennai (Jan 2002).

3. Invited speaker at the Quantum Gravity workshop held at RAC centre, Ooty (Oct 2001).

4. Invited to give a seminar at T.I.F.R (Aug 2001).

5. Gave seminars at Caltech, Harvard, Princeton , Rutgers Universities, May 2001.

6. Visited ICTP, Trieste, Italy during May-June 2003 under Junior associate fellowship and

gave a seminar on my research work.

7. Invited to give a seminar at ULB, Brussels during May 2003.

8. Invited to University of Nantes, France to give a seminar.

9. Invited speaker for the Simons Workshop on String theory held at University of Stony-

brook, USA (during August 2003).

10. Invited speaker at the Topology, Geometry Workshop held at Univ. Of Cochin, Dec 2003.

11. Visited Perimeter Institute, Waterloo during June-July 2007 and gave a seminar on my

work on ‘Chern-Simons theory and its connections to Topological strings.

12. Visited CERN theory division, Geneva during May-June 2008 and presented my work on

“ Chern-Simons field theoretic invariants and topological strings.”

13. Invited speaker at the Indian Strings Meeting held at Pondicherry (Dec 6th -13th 2008).

I gave a talk on “ Dimer Models and Quiver Gauge Theories”.

14. Invited speaker at the knot theory conference held at ICTP (May 2009). I gave a talk on

“Detection of chirality and mutation of knots and links.”

15. Invited speaker at the Quantum Theory and Symmetries conference(QTS06) held at Uni-

versity of Kentucky (July 2009).

16. Invited speaker at the String Theory and Cosmology workshop held at Busan, S.Korea

(June 2011). I gave a talk on ‘Chern-Simons Theory and Topological Strings.”

17. Chaired a session in the ‘Mathematics and Applications of Branes in String Theory and

M-Theory held at Isaac Newton Institute of mathematical sciences, Cambridge, UK (Jan

10-15, 2012).

18. Invited speaker at the N=2 conference held at Mcgill University, Montreal (June 2012).

19. Invited to give three lectures in the SMS meeting held at University of Montreal during

June-July 2013.

AWARDS AND RECOGNITION

1. Merit Scholarship, IIT Madras, 1989-91.

2. Member of NOSC (National Organising String Committee), STRINGS 2001.

3. Recipient of IIT Bombay Research paper award for the year 2006.

The details of the paper:

On Link Invariants and Topological String Amplitudes

P. Ramadevi and Tapobrata Sarkar,

Journal Nuclear Physics B 600 [PM] (2001), 487-511

Highlights from reviewer comments:

A Nobel Laureate who reviewed the paper had this to say:

impressive calculation that did introduce some new technical tools,

characterized by high rigor and precision,

paper had a useful role in establishing the credibility of the Ooguri-Vafa conjecture

Leadership roles

1. Member of SERC school (High-energy Physics) planning committee

2. Presently, expert committee member for DST - Women’s scientist scheme selection.

Publications

• Papers in International Journals

1. P. Ramadevi, T.R. Govindarajan and R.K. Kaul, “ Chern-Simons theory as a theory

of knots and links III: compact semi-simple group,” Nucl. Phys. B 402 (1993) 548-

566.

2. P. Ramadevi, T.R. Govindarajan and R.K. Kaul, “ Knot Invariants from rational

conformal field theories,” Nucl. Phys. B 422 (1994) 291-306.

3. P. Ramadevi, T.R. Govindarajan and R.K. Kaul, “ Chirality of knots 942 and 1071

and Chern-Simons theory,” Mod. Phys. Lett. A 9,No.34 (1994) 3205-3218.

4. B. Basu-Mallick and, P. Ramadevi, “ Construction of Yangian algebra through a

multideformation parameter dependent rational R-matrix,” Phys. Lett. A 211,

(1996) 339-344.

5. P. Ramadevi, T.R. Govindarajan and R.K. Kaul, “ Representations of Composite

Braids and Invariants for Mutant Knots, and Links in Chern-Simons field theories,”

Mod. Phys. Lett. A 10 (1995) 1635-1658.

6. B. Basu-Mallick, P. Ramadevi, R. Jagannathan, “ Multiparametric and coloured

extensions of the quantum group GLq(N)and the Yangian Algebra Y(glN ) through

a symmetry transformation of the Yang-Baxter equation,” Int. Jour. Mod. Phys.

A, 12, No.5 (1997) 945-962.

7. Saurya Das, Arundhati Dasgupta and P. Ramadevi, “ Can extremal black holes have

non-zero entropy ?” Mod. Phys. Lett. A 12 (1997) 3067-3080.

8. Atish Dabholkar, Gautam Mandal, P. Ramadevi, “Nonrenormalisation of mass of

some non-supersymmetric states,” Nucl. Phys. B 520 (1998) 117-131.

9. I.P. Ennes, P. Ramadevi, A.V. Ramallo, J.M. Sanchez de Santos, “Duality in OSP (1|2)

conformal field theory and link invariants,” Int. Jour. Mod. Phys. A. 13 (1998)

2931-2978.

10. Sumit R. Das, Samir D. Mathur, S. Kalyana Rama, P. Ramadevi, “Boosts, Schwarzschild

Black Holes and Absorption cross-sections in M theory,” Nucl. Phys. B 527 (1998)

187-204.

11. Saurya Das, Arundhati Dasgupta, P. Ramadevi, Tapobrata Sarkar, “Planckian Scat-

tering of D-branes,” Phys. Lett. B 428 (1998) 51-58.

12. Sumit R. Das, Samir D. Mathur, P. Ramadevi, “Hawking Radiation from Four-

dimensional Schwarschild Black Holes in M-theory,” Phys. Rev. D 59 (1999) 084001.

13. P. Ramadevi and Swatee Naik, “ Computation of Lickorish’s Three manifold invari-

ants using Chern-Simons Theory,” Commun. Math. Phys. 209 (2000) 29-49.

14. P. Ramadevi, “Supergravity Solution for Three-String Junction in M-theory,” JHEP

0006:005, 2000.

15. R.K. Kaul and P. Ramadevi, “ Three-Manifold Invariants from Chern-Simons Field

Theory with Arbitrary Semi-Simple Gauge Groups,” Commun. Math. Phys. 217

(2001) 295-314.

16. P. Ramadevi and, Tapobrata Sarkar, “ On Link Invariants and Topological String

Amplitudes,” Nucl. Phys.B 600 (2001) 487-511.

17. S.Das, P. Ramadevi, U.A. Yajnik, “ Black Hole Area Quantization,” Mod. Phys.

Lett. 17(2002) 993.

18. S.Das, P. Ramadevi, U.A. Yajnik, A. Sule, “Quantum Mechanical Spectra of Charged

Black Holes,” Phys. Lett. B565(2003) 201-206.

19. N. Ananthkrishnan and P.Ramadevi, “Consistent Approximations to Aircraft Lon-

gitudinal Modes,” Journal of Guidance 25 Engineering Notes (2003)820-824.

20. N. Ananthkrishnan; P. Ramadevi, “Reply by the Authors to G. Mengali,” Journal

of Guidance, Control, and Dynamics 2003, 0731-5090 vol.26 no.2 (383-383).

21. Pravina Borhade, P. Ramadevi, Tapobrata Sarkar, “ U(N) Framed Links, Three-

Manifold Invariants, and Topological Strings,” Nucl.Phys. B678(2004)656-681.

22. Pravina Borhade, P. Ramadevi, “SO(N) Reformulated Link Invariants from Topo-

logical Strings,” Nucl.Phys. B727 (2005) 471-498.

23. Saurya Das, Himan Mukhopadhyay and P. Ramadevi, “The Spectrum of rotating

black holes and its implications & for Hawking radiation,” Class. Quantum Grav.

22 (2005) 453-465.

24. R.K. Kaul, T.R. Govindarajan and P.Ramadevi, “Schwarz-Type Topological Quan-

tum Theories,” hepth-0504100, Encyclopedia of Mathematical Physics 494-503, World

Scientific May 2006 issue.

25. Pravina Borhade and P.Ramadevi, “Effective SO superpotential for N=1 theory with

Nf fundamental matter,” Nuclear Physics B 774 (2007) 323-339.

26. Prarit Agarwal, P. Ramadevi, Tapobrata Sarkar, “ A note on dimer models and

D-brane gauge theories,” JHEP06(2008)054.

27. Chandrima Paul, Pravina Borhade, P. Ramadevi, “ Composite Representation In-

variants and Unoriented Topological String Amplitudes,” Nucl. Phys. B841 (2010)

448-462.

28. Siddharth Dwivedi and Pichai Ramadevi, “Inverse algorithm and M2-brane theo-

ries,” JHEP 1111 (2011) 111; arXiv 1108.2387[hep-th]

29. Satoshi Nawata, P. Ramadevi, Zodinmawia,Xinyu Sun, “ Super-A-polynomials for

Twist Knots,” JHEP 1211 (2012) 157.

30. Zodinmawia , P. Ramadevi , “ SU(N) quantum Racah coefficients & non-torus links,”

Nucl.Phys. B870 (2013) 205-242.

31. Siddharth Dwivedi, P. Ramadevi ,“ Partial resolution of complex cones over Fano

B,”Adv.High Energy Phys. 2013 (2013) 295842.

32. Satoshi Nawata, P. Ramadevi, Zodinmawia, “Multiplicity-free quantum 6j-symbols

for Uq(slN ),” Lett.Math.Phys. 103 (2013) 1389-1398

33. Satoshi Nawata, P. Ramadevi, Zodinmawia, “Colored HOMFLY polynomials from

Chern-Simons theory,”J. of Knot theory and its Ramifications. vol. 22, No.13(2013)1350078;

e-Print: arXiv:1302.5144 [hep-th]

34. Satoshi Nawata, P. Ramadevi, Zodinmawia,“Colored Kauffman Homology and Super-

A-polynomials,” JHEP 01 (2014) 126; e-print:arXiv:1310.2240 (hep-th)

• Paper in arXiv

1. Siddharth Dwivedi, P. Ramadevi ,“ Is toric duality a Seiberg-like duality in (2+1)-d

?,” eprint: arXiv:1401.2767(hep-th)

2. Satoshi Nawata, P. Ramadevi, Zodinmawia, “ Trivalent graphs, volume conjectures

and character varieties,” e-Print: arXiv:1404.5119 [math.GT]

• Popular Articles in National Journals

1. P. Ramadevi,“Exchange of Identical Particles,” Resonance Journal,Feb 2001, pg 23-

28.

2. Akshay Kulkarni, P. Ramadevi, “Supersymmetry,” Resonance Journal, Feb 2003, pg

28-41.

3. P. Ramadevi, “String Theory- A pedagogical Review,” Physics News Jan 2010 issue.

4. Rajendra Shinde, Sushant Raut, P. Ramadevi, “Random matrices- An approach to

understand complex systems,” Resonance Journal, August 2011, pg722-741.

• Conference Proceedings

Pichai Ramadevi, “Detection of Chirality and Mutations of Knots and Links,” an

article in the Introductory Lectures on Knot Theory (Selected Lectures Presented at

the Advanced School and Conference on Knot Theory and its Applications to Physics

and Biology), Series on Knots and Everything vol. 46, World Scientific 2012, edited

by L.H. Kauffman, Sofia Lambropoulou, Slavik Jabllan and Jozef H Przytycki.

• Books Published

1. P. Ramadevi, Editor, Surveys In Theoretical High Energy Physics 1(Lecture Notes

from SERC Schools), Texts and Readings in the Physical Sciences-12, Hindustan

Book Agency.

Nuclear Physics B 678 [PM] (2004) 656–681

www.elsevier.com/locate/npe

U(N) framed links, three-manifold invariants,and topological strings

Pravina Borhadea, P. Ramadevia, Tapobrata Sarkarb

a Department of Physics, Indian Institute of Technology Bombay, Mumbai 400 076, Indiab Abdus Salam International Centre for Theoretical Physics, Strada Costiera, 11-34014 Trieste, Italy

Received 12 August 2003; received in revised form 2 October 2003; accepted 11 November 2003

Abstract

Three-manifolds can be obtained through surgery of framed links inS3. We study the meaning ofsurgery procedures in the context of topological strings. We obtainU(N) three-manifold invariantsfrom U(N) framed link invariants in Chern–Simons theory onS3. These three-manifold invariantsare proportional to the Chern–Simons partition function on the respective three-manifolds. Using thetopological string duality conjecture, we show that the largeN expansion ofU(N) Chern–Simonsfree energies on three-manifolds, obtained from some class of framed links, have a closed stringexpansion. These expansions resemble the closed string A-model partition functions on Calabi–Yaumanifolds with one Kahler parameter. We also determine Gopakumar–Vafa integer coefficients andGromov–Witten rational coefficients corresponding to Chern–Simons free energies on some three-manifolds. 2003 Elsevier B.V. All rights reserved.

PACS: 11.25.-w; 11.15.Pg

1. Introduction

After the second superstring revolution, several useful relations have been discoveredunifying various ideas of physics and mathematics. One such surprising discovery in therecent past has been the new connections between Chern Simons gauge theory and thephysics of closed topological string theory in certain backgrounds.

E-mail addresses: [email protected] (P. Borhade), [email protected] (P. Ramadevi),[email protected] (T. Sarkar).

0550-3213/$ – see front matter 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2003.11.023

http://www.elsevier.com/locate/npe

P. Borhade et al. / Nuclear Physics B 678 [PM] (2004) 656–681 657

The initial steps in this direction was taken by Gopakumar and Vafa in [1–3]. Theconjecture put forward by these authors relate largeN Chern–Simons gauge theory onS3,which is equivalent to A-twisted open topological string theory onT ∗S3 [4], to the A-typeclosed topological string theory on the resolved conifold. This conjecture was then tested atthe level of the observables of the Chern Simons theory, namely the knot invariants. In [5],Ooguri and Vafa formulated the conjecture in terms of invariants for the unknot (a circle inS3), and further checks were carried out for more nontrivial knots in [6–9].

The evaluation of knot invariants in [5] actually led to very strong integrality predictionsfor the (instanton generated) A-model disc amplitudes, which were then verified from themore tractable mirror B-model side by several authors [10–13].

Purely from gauge theory considerations, following the idea of ’t Hooft [14], it looks tobe a challenging problems to prove that the Feynmann perturbative expansion of anyU(N)

gauge theory in the largeN limit is equivalent to a closed string theory. It is believed thatthe Gopakumar–Vafa duality conjecture can provide insight in determining the ’t Hooftexpansion ofU(N) Chern–Simons free energy on any three-manifoldM.

As we have already mentioned, the Gopakumar–Vafa duality conjecture states thatU(N) Chern–Simons theory onS3, which describes the topological A-model ofND-branes onX = T ∗S3, is dual to topological closed string theory onXt = O(−1) ⊕O(−1) → P1. Having verified the conjecture at the level of Chern–Simons partitionfunction onS3 and Wilson loop observables (the knot invariants), we need to understandthe meaning of surgery of framed links inS3 within the context of topological strings, andthis is one of the issues we set out to address in this paper.

From the fundamental theorem of Lickorish and Wallace [15], it is well known thatany three-manifoldM can be obtained by surgery on a framed link inS3. Further, twoframed links related by a set of moves called Kirby moves determine the same manifold.In Chern–Simons theory, an algebraic expression has been derived [16], in terms of framedlink invariants, which are unchanged under Kirby moves of the framed links. Hencethe algebraic expression represents three-manifold invariants which are proportional tothe Chern–Simons partition function (Z[M]) on the three-manifoldM. Incorporating theresults of the topological string duality conjecture, we determine largeN expansion forlnZ[M] for many manifolds. Surprisingly, the expansion looks like an A-model closedstring partition function on a Calabi–Yau space with one Kahler parameter.

We point out an important subtelty here. As is well known, the classical solutions ofthe Chern–Simons action on a general three-manifoldM are the flat connections onM.In the weak coupling limit, the Chern–Simons partition function gets contributions froma perturbative expansion around all such stationary points. Noting that the space of flatconnections may be either a collection of a set of stationary points or a set of connectedpieces, this partition function can be appropriately written as a sum or an integral over thespace of flat connections. The largeN expansion of ’t Hooft is expected to relate the 1/N

expansion of the Chern–Simons theory around a given flat connection to an A-type closedtopological string theory. In [17], this has been shown from a matrix model approach,for the Lens spaceL(p,1). In this paper, however, we show that the full Chern–Simonspartition function(lnZ[M]) has a closed string interpretation for a class of three manifolds.

Indeed, proposing new duality conjectures between Chern–Simons theory on generalthree-manifoldsM and the corresponding dual closed string theories will involve the

658 P. Borhade et al. / Nuclear Physics B 678 [PM] (2004) 656–681

extraction of the partition function around individual flat connections, in lines with [17].This will extend the original conjecture by Gopakumar and Vafa, for general manifoldsM.We believe that our results on the invariants, involving the full partition function, would beuseful in proposing and understanding fully the nature of such dualities. We will elaborateon this point further in the concluding section.

The organisation of the paper is as follows. In Section 2, we briefly recapitulatethe framed link invariants inU(N) Chern–Simons theory, and present theU(N) three-manifold invariants obtained from framed link invariants inS3. In Section 3, we show therelation between the three-manifold invariants and the observables in topological stringtheory. Further, we obtain closed string invariants for the Chern–Simons free energies.Section 4 contains some explicit results on the Gopakumar–Vafa coefficients correspondingto the largeN expansion of the Chern–Simons free energy on some manifolds. Section 5ends with some discussions and scope for future research. In Appendix A, we present someresults on the integer invariants for the unknot with arbitrary framing, which are useful forthe computation of the Gopakumar–Vafa coefficients.

2. U(N) Chern–Simons gauge theory

Chern–Simons gauge theory on a three-manifoldM based on the gauge groupU(N) isa factored Chern–Simons theory of two gauge groups,SU(N) andU(1). That is, the actionis simply a sum of two Chern–Simons actions, one for gauge groupSU(N) and the otherfor U(1), each with an independent coupling constant (k, k1)

(2.1)S = k

4π

∫M

Tr

(A∧ dA+ 2

3A∧A∧A

)+ k1

4π

∫M

Tr(B ∧ dB),

whereA is a gauge connection for gauge groupSU(N) and B is the connection forU(1). Clearly, theU(N) partition functionZ{U(N)}[M] ≡Z[M] is just the product of twopartition functions [Z{SU(N)}[M], Z{U(1)}[M]]

(2.2)Z[M] =∫

[DB][DA]eiS.

We shall now briefly present the Wilson loop observables in the theory. TheU(N)

Wilson loop operators for ar-component linkL made up of component knotsKi ’s aresimply factored Wilson operators of theU(1) andSU(N) theories

(2.3)W{(Ri ,ni )}[L] =r∏i=1

TrRi U(A)[Ki]Trni U

(B)[Ki],

whereU(A)[Ki] = P [exp∮KiA] denotes the holonomy of theSU(N) gauge fieldA

around the component knotKi of a link L carrying representationRi andU(B)[Ki ] =P [exp

∮KiB] denotes the holonomy of theU(1) gauge fieldB around the component knot

Ki carryingU(1) chargeni . The expectation value of these Wilson loop operators are the


U(N) link invariants which are products ofSU(N) andU(1) invariants

V{U(N)}(R1,n1),(R2,n2),...,(Rr ,nr )

[L,M]

= ⟨W{(Ri ,ni )}[L]⟩= ∫ [DA][DB]eiSW{(Ri ,ni)}[L]∫ [DA][DB]eiS

(2.4)= V{SU(N)}R1,R2,...,Rr

[L,M]V {U(1)}n1,n2,...,nr

[L,M].

2.1. U(N) framed link invariants in S3

The observables inU(1) Chern–Simons theory on a three-sphereS3 capture only self-linking numbers (also called framing numbers) and the linking numbers between thecomponent knots of any link. Hence, theU(1) link invariant will be

(2.5)V {U(1)}n1,n2,...,nr

[L,S3]= exp

(iπ

k1

r∑i=1

n2i pi

)exp

(iπ

k1

∑i =j

ninj �kij

),

wherepi ’s are the framing numbers of the component knotsKi ’s and�kij are the linkingnumbers between the component knotsKi ,Kj . From Eqs. (2.4), (2.5), it is clear that theU(N) link invariants coincides withSU(N) invariants if and only ifpi ’s and�kij are zero.

The evaluation ofSU(N) framed link invariants from Chern–Simons theory onS3

makes use of the two ingredients: (i) the connection between Chern–Simons field theoryand the corresponding Wess–Zumino conformal field theory, (ii) the fact that knots andlinks can be obtained by closure or platting of braids. We refer the reader to [16,18]for detailed description of obtainingSU(2) framed invariants and framed invariants fromChern–Simons theory onS3 based on any arbitrary semi-simple group.

We shall now present the polynomials for various framed knots and links. For the unknotU with an arbitrary framingp, carrying a representationR of SU(N), the polynomial is

(2.6)V{SU(N)}R

[0(p), S3]= q(pCR)VR[U ] = q− p�2

2N(qpκRdimqR

),

whereq = exp( 2πik+N ), � refers to the total number of boxes in the Young tableau of the

representationR and

(2.7)κR = 1

2

(N�+ �+

∑i

(l2i − 2ili

)),

with li being the number of boxes in theith row of the Young tableau of therepresentationR. One can verify that both the quantum dimension of a representation,

dimqR and κR are polynomials in variablesλ± 12 and q± 1

2 whereλ = qN . The frame

dependent term involves a variableez = q1

2N . Hence the unknot invariants with framing

p = 0 are no longer polynomials in variablesq± 12 , λ± 1

2 but also involve one more variable

ez = q1

2N .


In order to make the polynomials independent of the variableez, we can multiply byU(1) invariant (2.5) with a specific choice ofU(1) chargen and coupling constantk1

(2.8)n= �√N, k1 = k +N.

Therefore theU(N) invariant for thep-framed unknot inS3 with the above choice ofU(1)representation and coupling constant is

(2.9)V{U(N)}(R, �√

N)

[0(p), S3]= qpκRdimqR.

Now, we can write theU(N) framed knot invariants for torus knots of the type(2,2m+ 1)and other nontorus knots like 41, 61 in S3. For example, theU(N) invariant for a framedtorus knot of typeK ≡ (2,2m+ 1) with framing[p− (2m+ 1)] will be

(2.10)V{U(N)}(R, �√

N)

[K,S3]= qpκR

∑Rs∈R⊗R

dimqRs(−1)εs(qκR−κRs /2)2m+1

,

whereεs = ±1 depending upon whether the representationRs appears symmetricially orantisymmetrically with respect to the tensor productR⊗R in theSU(N)k Wess–Zumino–Witten model. Similarly,U(N) invariants for framed torus links of the type(2,2m) canalso be written. For example, theU(N) invariant for a Hopf link with linking number−1and framing numbersp1 andp2 on the component knots carrying representionsR1 andR2will be

V{U(N)}(R1,

�1√N),(R2,

�2√N)

[H ∗(p1,p2), S

3](2.11)= qp1κR1qp2κR2

∑Rs∈R1⊗R2

dimqRsqκR1+κR2−κRs ,

where�1 and�2 refers to total number of boxes in the Young tableau of the representationsR1 andR2, respectively. From now on, we shall denote

V{U(N)}(R1,

�1√N),(R2,

�2√N),...,(Rr ,

�r√N)

[L,S3]≡ V

{U(N)}R1,R2,...,Rr

[L,S3]

suppressing theU(1) charges as they are related to the total number of boxes in theYoung tableau of the representationsRi ’s (2.8). Using the framedSU(N) invariants forarbitrary framed links inS3 [16], it is straightforward to obtain the correspondingU(N)link invariants. We will now see how theseU(N) framed invariants inS3 with such specialU(1) representation reflects on three-manifold invariants.

2.2. U(N) three-manifold invariants

The Lickorish–Wallace theorem states that any three-manifoldM can be obtained bya surgery of framed knots and links inS3. Two framed links related by Kirby moveswill determine the same three-manifold. In other words, three-manifold invariants mustbe constructed from framed link invariants in such a way that they are preserved underKirby moves. TheSU(N) three-manifold invariantF [M] for a manifoldM obtained by


surgery of a framed link inS3 will be [16]

(2.12)F [M] = α−σ [L] ∑R1,R2,...,Rr

µR1,R2,...,Rr

({pi}, {�kij })V {SU(N)}R1,R2,...,Rr

[L,S3],

where{pi}, {�kij } are the framing and linking numbers andσ(L) is the signature of thelinking matrix of framed linkL. It has been proven [16] thatF [M] is unchanged under theoperation of Kirby moves on framed links if we choose

(2.13)α = exp

(iπc

4

), µR1,R2,...,Rr

({pi}, {�kij })= S0R1S0R2 · · ·S0Rr ,

wherec = k(N2−1)(k+N) andS0Ri ’s denotes the modular transformation matrix elements. We

see thatµR1,R2,...,Rr is independent of framing and linking numbers. We can now constructU(N) three-manifold invariants from theU(N) framed link invariants inS3 as follows

(2.14)F [M] = β−σ [L]∑{Ri }

µR1,R2,...,Rr

({pi}, {�kij })V {U(N)}R1,R2,...,Rr

[L,S3],

whereβ and µ must be chosen such thatF [M] is unchanged under Kirby moves onframed links. Further, for obtaining three-manifolds from knots and disjoint links withzero framing and linking numbers, we require

(2.15)F [M] = F [M], as V{U(N)}R1,...,Rr

[L,S3]= V

{SU(N)}R1,...,Rr

[L,S3].

Therefore, for{pi} = 0, {�kij } = 0

(2.16)µR1,R2,...,Rr

({pi = 0}, {�kij = 0})= µR1,R2,...,Rr .

The special choice of theU(1) representation and the above limiting conditions suggeststhat

(2.17)β = α, µR1,R2,...,Rr

({pi}, {�kij })= µR1,R2,...,Rr e−z(∑i l

2i pi+

∑i =j li lj �kij

),

for F [M] to be preserved under Kirby moves. Rewriting theU(N) link invariants in termsof the SU(N) invariants, it is obvious thatF [M] is the same asF [M]. Hence the three-manifold invariants do not distinguish betweenU(N) andSU(N) gauge groups and theyare proportional to the partition functionZ[M] [16]

(2.18)F [M] = Z[M]Z[S3] .

The partition function onS3 is equal to

(2.19)Z[S3]= S00.

Let us introduce a slight modification in notation which will be useful when we relate thesethree-manifold invariants with expectation values of topological operators in topologicalstring theory

(2.20)V{U(N)}R1,R2,...,Rr

[L,S3]= (−1)

∑i �ipi λ

∑i �i pi2 V

{U(N)}R1,R2,...,Rr

[L,S3](q,λ).


TheU(N) (or equivalentlySU(N)) three-manifold invariants can be rewritten as

(2.21)Z[M]S00

= Z0[M]S00

+F[M,z, {pi}, {�kij }

],

whereZ0[M]/S00 is independent ofz, {pi}, {�kij } and is given by

Z0[M]S00

= α−σ [L] ∑R1,R2,...,Rr

S0R1 · · ·S0Rr

(2.22)× (−1)∑i �ipi λ

∑i �i pi2 V

{U(N)}R1,R2,...,Rr

[L,S3](q,λ),

andF [M,z, {pi}, {�kij }] contains the remainingz, {pi}, {�kij } dependent terms. For knotsand disjoint links with zero framing and linking numbers, it is not difficult to see thatF [M,z, {pi}, {�kij }] = 0 resulting inZ[M] = Z0[M]. In the next section, we will showthe natural appearance ofZ0[M] in the context of topological strings.

3. Topological strings

Gopakumar and Vafa have conjectured that closed topological string theory on aresolved conifold is dual to largeN Chern–Simons gauge theory onS3. The conjecture hasbeen verified by comparing the largeN expansion of the free-energy of the Chern–Simonstheory onS3 with the closed topological string amplitude near the resolved conifold. Thisduality relates the Chern–Simons field theory variablesq and λ with the string theoryparameters

(3.1)q = egs , λ= et = eNgs ,

wheregs is the string coupling constant andt is the Kahler parameter of the resolvedconifold. With the above identification between the variablesq,λ with gs, t , the Chern–Simons variableez is

(3.2)ez = egs/2N.

The largeN expansion is performed by taking the limits

(3.3)gs → 0 and N → ∞.

In this limit, the variablez= gs/2N can be set to zero. This suggests that thez independentpart of the three-manifold invariant, namely,Z0[M]/S00 can be compared with quantitieson the topological string side.

Ooguri and Vafa found another piece of evidence for this duality conjecture by showingthat the Wilson loop operators in Chern–Simons theory correspond to certain observablesin the topological string theory. The operators in the open topological string theory whichcontains information about links is given by [5]

(3.4)Z({Uα}, {Vα})= exp

[r∑

α=1

∞∑d=1

1

dTrUdα TrV dα

],


whereUα is the holonomy of the gauge connectionA around the component knotKαcarrying the fundamental representation in theU(N) Chern–Simons theory onS3, andVα is the holonomy of a gauge fieldA around the same component knot carrying thefundamental representation in theU(M) Chern–Simons theory on a Lagrangian three-cycle which intersectsS3 along the curveKα .

We can use some group theoretic properties to show that the expectation value of theoperator (3.4) exactly matchesZ0[M]/S00 provided we choose the rank of the two Chern–Simons gauge groups to be same (N =M). If we expand the exponential in Eq. (3.4), wewill get

(3.5)Z({Uα}, {Vα})= 1+

∑{�k(α)}

r∏α=1

1

z�k(α)γ�k(α) (Uα)γ�k(α) (Vα),

where

(3.6)z�k(α) =∏j

k(α)j !jk(α)j , γ�k(α) (Uα)=

∞∏j=1

(TrUjα

)k(α)j .

Here�k(α) = (k(α)1 , k

(α)2 , . . .) with |�k(α)| =∑

j k(α)j and the sum is over all the vectors�k(α)

such that∑rα=1 |�k(α)|> 0. Using the group theoretic properties

(3.7)γk1(U1) · · ·γkr (Ur)=∑

R1,...,Rr

r∏α=1

χRα(C(�k(α)))TrR1(U1) · · ·TrRr (Ur),

(3.8)∑

�k

1

z�kχR1

(C(�k))χR2

(C(�k))= δR1R2,

whereχRα(C(�k(α)))’s are characters of the symmetry groupS�α with �α =∑j jk

(α)j and

C(�k(α)) are the conjugacy classes associated to�k(α)’s (denotingk(α)j cycles of lengthj ),one can show that Eq. (3.5) becomes

(3.9)Z({Uα}, {Vα})=

∑{Rα}

r∏α=1

TrRα (Uα)TrRα (Vα).

Ooguri and Vafa have conjectured a specific form for the vacuum expectation value (vev)of the topological operators (3.4) for knots [5] invoking the largeN topological stringduality. This result was further refined for links [8,19] which is generalisable for framedlinks as follows

(3.10)⟨Z({Uα}, {Vα})⟩A = exp

[ ∞∑d=1

∑{Rα}

1

df(R1,...,Rr )

(qd,λd

) r∏α=1

TrRα Vdα

],

(3.11)f(R1,R2,...,Rr )(q, λ)= λ12

∑α �αpα

∑Q,s

1

(q1/2 − q−1/2)N(R1,...,Rr ),Q,sq

sλQ,


where the suffixA on the vev implies that the expectation value is obtained by integratingthe U(N) gauge fieldsA’s on S3, and �α is the total number of boxes in the Youngtableau of the representationRα . Further, for framed linksN(R1,...,Rr ),Q,s are integers onlyif expectation value of theU(N) Wilson loop operators inS3 are defined as [9]

(3.12)

⟨r∏

α=1

TrRα (Uα)

⟩= (−1)

∑rα=1 �αpαV

{U(N)}R1,...,Rr

[L,S3]

(3.13)= λ∑α�αpα

2 V{U(N)}R1,...,Rr

[L,S3](q,λ).

This is justified since the holonomyVα on the Lagrangian three-cycleC, under change offraming becomesVα = (−1)pαVα [9], which is equivalent to

(3.14)TrRα Vα = (−1)�αpα TrRα Vα.

If we also integrate theA fields in the Chern–Simons field theory on the Lagrangian three-cycle then vev of the topological operator (3.9) is

(3.15)⟨Z({Uα}, {Vα})⟩A,A =

∑R1,...,Rr

⟨r∏

α=1

TrRα Vα

⟩⟨r∏

α=1

TrRα Uα

⟩,

where we need to determine the expectation value of the Wilson loop operators on theLagrangian three-cycleC with Betti numberb1 = r which are noncompact. For thesenoncompact Lagrangian three-cycles, it appears to be possible to deform knots and linksinto unknot and disjoint collection of unknots, respectively. Therefore, it is convincing toassume

(3.16)〈TrRα Vα〉 = dimqRα ≡ S0Rα

S00.

Even though the assumption looks logical, finding a proof still remains a challengingquestion. Substituting Eqs. (3.13), (3.16) in (3.15) and comparing with Eqs. (2.22), (2.20),we get the relation

(3.17)ασ [L] Z0[M](S00)r+1

= ⟨Z({Uα}, {Vα})⟩A,A.

Also, from Eq. (3.10), we get

(3.18)ln

(ασ [L] Z0[M]

(S00)r+1

)=

∞∑d=1

∑R1,...,Rr

1

df(R1,...,Rr )

(qd,λd

) r∏α=1

⟨TrRα V

dα

⟩.

In the limit whenN → ∞ andgs → 0, the Chern–Simons partition functionZ[M] canbe approximated toZ0[M] (2.22) and the above equation relates the Chern–Simons freeenergy to the expectation value of the topological link operator (3.4) in topological closedstring theory. It is appropriate to mention that the information of other three-manifoldsobtained from surgery on framed links inS3 in Chern–Simons theory is captured byintegrating both theA and A gauge fields in the topological string theory. It will beinteresting to see whether the Chern–Simons free energy(lnZ0[M]) can be shown to be


equal to the closed string partition function in these cases. We will see later that the Chern–Simons free energy in fact resembles the A-model partition function.

Before proceeding to the largeN expansion of the free energy, we briefly recapitulatesome salient features of the A-model topological string partition function on Calabi–Yaumanifolds.

3.1. The A-model topological string partition function

The A-model topological partition function on a Calabi–Yau manifoldX with somenumber of Kahler parameters denoted byti ’s is defined as

(3.19)F (X)=∑g

g2g−2s Fg

({ti}),(3.20)Fg

({ti})=∑

{βi }∈H2(X,Z)

Ng{βi }e

− �β·�t ,

whereFg({ti}) are the A-model topological string amplitudes at genusg andNg

{βi} are theclosed string Gromov–Witten invariants associated to genusg curves in the homologyclass �β. In Ref. [3], a strong structure result has been derived from M-theory for thetopological string partition function

(3.21)F (X)=∑

{mi },r�0,d>0

1

dnr,{mi}

(2 sinh

dgs

2

)2r−2

exp[−d

(∑miti

)],

where nr,{mi } are integers usually referred to as Gopakumar–Vafa invariants. Clearly,the genusg Gromov–Witten invariantsNg

{βi } will involve the set of Gopakumar–Vafainvariantsnr�g,{mi}�{βi }.

We would like to derive a largeN closed string expansion for the Chern–Simons freeenergy lnZ0[M] on any three manifoldM (3.18) and show that it has the structure (3.21).It is not a priori clear whether the Chern–Simons free energy on anyM will have a closedstring interpretation. Interestingly, using the duality connection between Chern–Simonstheory onS3 and topological strings and some properties of group theory, we will showthat the free energy (3.18) does have the form (3.21) for a subset of knots and links.

We can write the RHS of Eq. (3.18) as follows

∑R1,...,Rr

f(R1,...,Rr )

(qd,λd

) r∏α=1

⟨TrRα V

dα

⟩(3.22)=

∑�k(1),...,�k(r)

f�k(1),...,�k(r)(qd,λd

) r∏α=1

1

z�k(α)⟨γ�k(α)

(V dα

)⟩,

wheref�k(1),...,�k(r) is the character transform offR1,...,Rr whose form has been derived in[8], namely,


f�k(1),...,�k(r)(q, λ)=(∏

j (qj2 − q− j

2 )∑rα=1 k

(α)j

(q12 − q− 1

2 )2

)(λ

12)[∑

α pα(∑

j jk(α)j

)]

(3.23)×∑Q

∑g�0

n(�k(1),...,�k(r)),g,Q

(q− 1

2 − q12)2gλQ,

where

(3.24)n(�k(1),...,�k(r)),g,Q =∑

R1,...,Rr

r∏α=1

χRα(C(�k(α)))N(R1,...,Rr ),g,Q.

N(R1,...,Rr ),g,Q are integers which compute the net number of BPS domain walls of chargeQ and sping transforming in the representationRα of U(M) in the topological stringtheory. AsVα ’s correspond to disjoint unknots with appropriate sign corresponding to theframing numbers, we can write

(3.25)

⟨r∏

α=1

γ�k(α)(V dα

)⟩= (−1)d[∑

α pα(∑

j jk(α)j

)]( ∏j (λ

dj2 − λ− dj

2 )

(qdj2 − q− dj

2 )

)∑rα=1 k

(α)j

.

Incorporating the above results in Eq. (3.18), we get

ln

(ασ [L] Z0[M]

(S00)r+1

)=

∞∑d=1

∑g,Q

1

d

(2 sinh

dgs

2

)2g−2

(3.26)

×{λdQ

∑�k(1),...,�k(r)

n(�k(1),...,�k(r)),g,Q

×r∏

α=1

(1

z�k(α)(−λ 1

2)dpα(∑j jk

(α)j

)∏j

(λ− dj

2 − λdj2)k(α)j

)}.

The RHS of the above equation has the structure of the free energy for a closed string(3.21) provided we can prove that the expression within parenthesis satisfies{

λdQ∑

�k(1),...,�k(r)n(�k(1),...,�k(r)),g,Q

×r∏

α=1

(1

z�k(α)(−λ 1

2)dpα(∑j jk

(α)j

)∏j

(λ− dj

2 − λdj2)k(α)j

)}

(3.27)=∑{mi }

ng,{mi}e−d∑i mi ti .

Identifying,λ= exp(t), we see that the above relation will be true for Calabi–Yau spaceswith one Kahler parameter after performing appropriate analytic continuation of the


variableλ→ λ−1 [5]. We should be able to extract the integer invariants (the Gopakumar–Vafa invariants) from the open string invariantsn(�k(1),...,�k(r)),g,Q. Using the results obtainedin Ref. [8], one can show that(−λ 1

2)dpα(∑j jk

(α)j

)∏j

(λ− dj

2 − λdj2)k(α)j

(3.28)= (λ− d

2 − λd2)∑Rα

(−λ 12)�αpαdχRα (C(�k(α)))SRα (λd),

whereSRα (λ) is zero ifRα is not a hook representation and ifRα is a hook representationwith �α boxes with�α − sα boxes in the first row (and we denote the representation asR�α,sα ) then we have

(3.29)SRα(λd)= (−1)sαλd(−

�α−12 +sα).

Using the properties (3.24), (3.28) in Eq. (3.26) we get

ln

(ασ [L] Z0[M]

(S00)r+1

)=

∞∑d=1

∑g

1

d

(2 sinh

dgs

2

)2g−2

×{∑

Q

∑{�α}

∑{sα}

N(R�1,s1 ,...,R�r ,sr ),g,Q(−1)

∑α sα (−1)d

∑α �αpαλ

12d

∑α �αpα

× (λd

{Q+∑α(− �α

2 +sα)} − λd{Q+1+∑α(− �α

2 +sα)})}

(3.30)=∑g,d,m

1

d

(2 sinh

dgs

2

)2g−2

ng,me−dmt .

We can obtain the Gopakumar–Vafa invariantsng,m, and also the Gromov–Witteninvariants, by evaluating theN(R1,...,Rr ),g,Q for various framed knots and links.

Before proceeding to outline the evaluation of theN ’s in the next subsection, weremind the reader that in our computation of the Gopakumar–Vafa and the Gromov–Witteninvariants, we use the full Chern–Simons partition function in (3.30), and show that thishas a closed string interpretation. For reasons that have been outlined in the introduction,our results do not constitute new dualities between Chern–Simons theory onM and closedA-type topological string theories, although we believe that these results would be veryimportant for gaining a full understanding of the same.

Also, note that even though one can check that∑α(−�α/2 +Q) is always an integer,

the term[(−1)λ1/2]∑α �αpα within the parenthesis of (3.30) can be integral powers ofλ

(for arbitrary�α ’s) if and only ifpα ’s on the components knots are even. This suggests thatthe closed string expansion (3.27) is possible only for framed knots and links with evennumbers of framing numberspα ’s on all the component knots.


3.2. Determination of the N ’s from framed link invariants

The general formula forf (3.11) in terms of framed link invariants (2.20) can be writtenas [19]

fR1,R2,...,Rr (q, λ)

= λ∑α �αpα/2

∞∑d,m=1

(−1)m−1µ(d)

dm

∑{�k(αj),Rαj }

r∏α=1

χRα

(C

((m∑j=1

�k(αj))d

))

(3.31)×m∏j=1

|C(�k(αj))|�αj ! χRαj

(C(�k(αj)))V {U(N)}

R1j ,R2j ,...,Rrj

[L,S3](qd,λd),

whereµ(d) is the Moebius function defined as follows: ifd has a prime decomposition({pi}), d = ∏a

i=1pmii , thenµ(d) = 0 if any of themi is greater than one. If allmi = 1,

thenµ(d) = (−1)a . The second sum in the above equation runs over all vectors�k(αj),with α = 1, . . . , r and j = 1, . . . ,m, such that

∑rα=1 |�k(αj)| > 0 for any j and over

representationsRαj . Further�kd is defined as follows:(�kd)di = ki and has zero entriesfor the other components. Therefore, if�k = (k1, k2, . . .), then

(3.32)�kd = (0, . . . ,0, k1,0, . . . ,0, k2,0, . . .),

wherek1 is in thed th entry,k2 in the 2d th entry, and so on. Hence, one can directly evaluatef from U(N) framed link invariants (2.20) and verify the conjecture (3.11). Using thefollowing equations

MR1,...,Rr ;R′1,...,R

′r=

∑R′′

1 ,...,R′′r

r∏α=1

CRαR′αR

′′αSR′′

α(q),

(3.33)

f(R′1,...,R

′r )(q, λ)= (

q−1/2 − q1/2)r−2 ∑g�0,Q

N(R′1,...,R

′r ),g,Q

(q−1/2 − q1/2)2gλQ,

we can write Eq. (3.31) as

(3.34)fR1,R2,...,Rr (q, λ)= λ12

∑α �αpα

∑R′

1,...,R′r

MR1,...,Rr ;R′1,...,R

′rf(R′

1,...,R′r )(q, λ).

In Eq. (3.33),Rα , R′α , R′′

α are representations of the symmetric groupS�α which can belabelled by a Young tableau with a total of�α boxes andCRR′R′′ are the Clebsch–Gordancoefficients of the symmetric group. In the next section, we will evaluateN for few framedknots, links and present the results of our computation of the Gopakumar–Vafa invariantsand Gromov–Witten invariants.


4. Examples and explicit results

Our aim in this section is to compute the Gopakumar–Vafa invariants and Gromov–Witten invariants corresponding to the Chern–Simons free energy on three manifoldsobtained from surgery of the respective framed links inS3.

4.1. Knots in standard framing

In standard framing, there is no distinction betweenU(N) andSU(N) knot invariants.Therefore for this class of knots with zero framing number, the Chern–Simons partitionfunction will be

(4.1)Z[M] = Z0[M],and the signature of the linking matrixσ [L] = 0. Substituting in Eq. (3.30), we get

lnZ[M] − 2 lnZ[S3]

=∞∑d=1

∑g

1

d

(2 sinh

dgs

2

)2g−2

×{∑

Q

∑�,s

N(R�,s ),g,Q(−1)s(λd(Q− �

2+s) − λd(Q+1− �2+s))}

(4.2)=∑g,d,m

1

d

(2 sinh

dgs

2

)2g−2

ng,me−dmt .

We will now compute the integer coefficientsng,m for few examples.

4.1.1. UnknotThe surgery of this simplest knot gives manifoldS2×S1 whose Chern–Simons partition

functionZ[S2 × S1] = 1. The nonzeroN ’s for the simplest unknot is [5]

(4.3)N ,0,Q=±1/2 = ∓1.

Substituting this result in Eq. (4.2) and doing the appropriate analytic continuationλ→λ−1, the nonzerong,m is

(4.4)n0,1 = 2.

Hence Eq. (4.2) reduces to

(4.5)−2 lnZ[S3]=

∑d

1

d(2 sinhdgs2

)2 2e−dt .

Clearly, the largeN expansion of Chern–Simons free-energy onS3 gives Gopakumar–Vafainvariant (ng,m[S3])

(4.6)n0,1[S3]= −1.


Table 1N

,g,Qfor the torus knot(2,5)

Q g = 0 g = 1 g = 2

3/2 3 4 15/2 −5 −5 −17/2 2 1 0

Table 2N


Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7

3 20 60 69 38 10 1 0 04 −80 −260 −336 −221 −78 −14 −1 05 120 400 534 366 136 26 2 06 −80 −260 −336 −221 −78 −14 −1 07 20 60 69 38 10 1 0 0

Table 3N


Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7

3 30 115 176 137 57 12 1 04 −120 −490 −819 −724 −365 −105 −16 −15 180 750 1286 1174 616 186 30 26 −120 −490 −819 −724 −365 −105 −16 −17 30 115 176 137 57 12 1 0

Using the integer invariants, we can evaluate the Gromov–Witten invariants

(4.7)N0m>0

[S3]= −1

m3 , N1m>0

[S3]= 1

12m, N2

m>0

[S3]= −m

240.

These invariants in the closed topological string partition function imply that the targetCalabi–Yau space is a resolved conifold.

4.1.2. Torus knots of type (2,2m+ 1)The knots obtained as a closure of two-strand braid with 2m+ 1 crossings are the type

(2,2m+ 1) torus knots. The surgery of these torus knots inS3 will give Seifert homologyspheresX( 2

−1,2m+1m+1 ,

−(2m+1)1 ). It will be interesting to determine the Gopakumar–Vafa

integer invariants and the closed Gromov–Witten invariants corresponding to largeN

expansion of Chern–Simons free-energy on such Seifert manifolds.(i) The NR,g,Q corresponding to the torus knot(2,5) for representations upto three

boxes are shown in Tables 1–6.The surgery of the torus knot(2,5) gives the Seifert manifoldM1 ≡ X( 2

−1,53,

−51 ).

Comparing powers ofλ−1 (after analytic continuation) in Eq. (4.2) we can obtainGopakumar–Vafa integer invariants (ng,m[M1]), corresponding to largeN expansion of


Table 4N


g Q= 9/2 Q= 11/2 Q= 13/2 Q= 15/2 Q= 17/2 Q= 19/2 Q= 21/2

0 232 −1652 4820 −7400 6320 −2852 5321 1436 −11626 37290 −61400 55140 −25726 48862 4046 −38060 135824 −241510 228824 −110390 212663 6781 −75590 303114 −584700 584729 −290550 562164 7384 −100086 456013 −958591 1011218 −514589 986515 5384 −92128 483836 −1115009 1240265 −642511 1201636 2636 −60064 370471 −943863 1107524 −580839 1041357 851 −27853 206727 −589169 730275 −385792 649618 173 −9107 83995 −272258 357280 −189269 291869 20 −2048 24548 −92689 129164 −68333 9338

10 1 −301 5020 −22898 34006 −17899 207111 0 −26 681 −3984 6331 −3304 30212 0 −1 55 −462 789 −407 2613 0 0 2 −32 59 −30 114 0 0 0 −1 2 −1 015 0 0 0 0 0 0 016 0 0 0 0 0 0 017 0 0 0 0 0 0 0

Table 5N


g Q= 9/2 Q= 11/2 Q= 13/2 Q= 15/2 Q= 17/2 Q= 19/2 Q= 21/2

0 778 −5483 15755 −23750 19880 −8783 16031 5929 −46514 145060 −232875 204460 −93539 174792 20986 −186222 636631 −1095250 1012006 −479592 914413 44960 −458386 1732046 −3205225 3116781 −1523901 2937254 64066 −764419 3219215 −6427475 6569210 −3295156 6345595 63300 −905137 4290087 −9275345 9951953 −5092334 9674766 44151 −780483 4213699 −9914620 11161326 −5796079 10720067 21814 −496526 3099128 −7990833 9440203 −4952673 8788878 7564 −233985 1719912 −4905169 6086843 −3213286 5381219 1795 −81304 720501 −2301847 3004880 −1590473 246448

10 277 −20526 226187 −823741 1133523 −599603 8388311 25 −3656 52319 −222684 323786 −170666 2087612 1 −435 8643 −44635 68760 −36018 368413 0 −31 964 −6421 10510 −5458 43614 0 −1 65 −626 1092 −561 3115 0 0 2 −37 69 −35 116 0 0 0 −1 2 −1 017 0 0 0 0 0 0 0

the Chern–Simons free energy onM1, in terms ofN ’s as follows (Table 7):

ng,1[M1] = N ,g,3/2 − 2δg,0,

ng,2[M1] = N ,g,5/2 − N ,g,3/2 + N ,g,3,


Table 6N


g Q= 9/2 Q= 11/2 Q= 13/2 Q= 15/2 Q= 17/2 Q= 19/2 Q= 21/2

0 612 −4282 12170 −18100 14920 −6482 11621 5506 −42271 129215 −203600 175690 −79121 145812 23192 −198031 657139 −1103915 1000889 −467586 883123 59384 −572273 2078169 −3736000 3559209 −1718033 3295444 101720 −1124627 4504027 −8688844 8685457 −4307675 8299425 121996 −1578123 7029420 −14594746 15285709 −7746221 14819656 104816 −1625710 8134469 −18248211 20007137 −10307863 19353627 65114 −1249175 7106311 −17316684 19871498 −10362997 18859338 29205 −721405 4735067 −12623787 15169790 −7976692 13878229 9339 −313241 2415866 −7114329 8962560 −4735426 775231

10 2071 −101529 941668 −3102690 4104397 −2172177 32826011 302 −24156 277825 −1042441 1450978 −766965 10445712 26 −4090 60907 −266857 391757 −206277 2453413 1 −466 9605 −51024 79211 −41446 411914 0 −32 1029 −7046 11600 −6018 46715 0 −1 67 −663 1161 −596 3216 0 0 2 −38 71 −36 117 0 0 0 −1 2 −1 0

Table 7ng,m[M1]m g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 g = 8 g = 9 g = 10 g � 11

1 1 4 1 0 0 0 0 0 0 0 0 02 12 51 67 38 10 1 0 0 0 0 0 03 109 1407 3466 6385 7239 5357 2634 851 173 20 1 0

ng,3[M1] = N ,g,7/2 − N ,g,5/2 − N ,g,3 + N ,g,4 − N ,g,3

(4.8)+ N ,g,9/2.

From these invariants, we can extract Gromov–Witten invariants which are rationalnumbers. A few of them are given in Table 8.

(ii) The NR,g,Q computation of the torus knot(2,7) will be shown in Tables 9–11.From Eq. (4.2), we can obtain Gopakumar–Vafa integer invariantsng,m[M2] corre-

sponding to Chern–Simons theory on Seifert manifoldM2 = X( 2−1,

74,

−71 ). We present

few of them in terms ofNR,g,Q

ng,1[M2] = −2δg,0,

ng,2[M2] = N ,g,5/2,

ng,3[M2] = N ,g,7/2 − N ,g,5/2,

ng,4[M2] = N ,g,9/2 − N ,g,7/2 + N ,g,5,

(4.9)ng,5[M2] = −N ,g,9/2 − N ,g,5 + N ,g,6 − N ,g,5.


Table 8Ngm[M1]

m g = 0 g = 1 g = 2

1 1 4712

241240

2 978

124724

8287120

Table 9N


Q g = 0 g = 1 g = 2 g = 3

5/2 −4 −10 −6 −17/2 7 14 7 19/2 −3 −4 −1 0

Table 10N


g Q= 5 Q= 6 Q= 7 Q= 8 Q= 9

0 −84 336 −504 336 −841 −574 2380 −3612 2380 −5742 −1652 7182 −11060 7182 −16523 −2623 12144 −19042 12144 −26234 −2529 12739 −20420 12739 −25295 −1536 8673 −14274 8673 −15366 −589 3892 −6606 3892 −5897 −138 1141 −2006 1141 −1388 −18 210 −384 210 −189 −1 22 −42 22 −1

10 0 1 −2 1 011 0 0 0 0 0

Table 11N


g Q= 5 Q= 6 Q= 7 Q= 8 Q= 9

0 −112 448 −672 448 −1121 −896 3696 −5600 3696 −8962 −3052 13104 −20104 13104 −30523 −5812 26300 −40976 26300 −58124 −6844 33188 −52688 33188 −68445 −5212 27692 −44960 27692 −52126 −2607 15640 −26066 15640 −26077 −849 6003 −10308 6003 −8498 −173 1541 −2736 1541 −1739 −20 253 −466 253 −20

10 −1 24 −46 24 −111 0 1 −2 1 0


From these integer invariants, it is straightforward to obtain Gromov–Witten rationalnumbers.

It appears from the computation ofNR,g,Q’s for the two torus knots that the rangeof Q is �(2m − 1)/2 � Q � �(2m + 3)/2. This range ofQ allows finite number ofN ’s to contribute tong,m,N

gm invariants. Thus we see that the largeN expansion of the

Chern–Simons free energy on Seifert manifolds obtained from surgery of torus knots oftype(2,2m+ 1) can be given a closed string interpretation. The target Calabi–Yau spaceY corresponding to the closed topological string theory must have the Gopakumar–Vafainteger invariants and closed Gromov–Witten invariants determined from Chern–Simonstheory. So far, we considered only knots in standard framing. In the next subsection weaddress three-manifolds obtained from framed knots inS3.

4.2. Framed knots

We have seen that the Chern–Simons partition functionZ[M], corresponding tomanifolds obtained from knots with nonzero framing number, can be approximated toZ0[M] in the limitN → ∞, gs → 0. We shall now determine lnZ0[M] for framed unknotwith even framing number 2p.

Unknot with framing p: the surgery ofp-framed unknot results in Lens spacesL(p,1).TheN ’s for the unknot with framingp = 4 are shown in Tables 12–17.

Using these NR,g,Q, we can determine Gopakumar–Vafa integer invariantsng,m[L(4,1)] from the expansion of lnZ0[L(p,1)]. We present some of the nonzero coef-ficients

ng,1[L(4,1)

]= N ,g,−1/2δg,0 − 2δg,0,

ng�1,2[L(4,1)

]= N ,g,1/2 − N ,g,−1/2 + N ,g,−1,

ng�4,3[L(4,1)

]= −N ,g,1/2 + N ,g,0 − N ,g,−1 − N ,g,−1 + N ,g,−3/2,

(4.10)

ng,4[L(4,1)

]= N ,g,1 − N ,g,0 − N ,g,0 + N ,g,−1 + N ,g,−1/2

− N ,g,−3/2 − N ,g,−3/2 + N ,g,−2,

Table 12N

,g,Qfor the unknot with framingp= 4

Q g = 0

−1/2 11/2 −1

Table 13N

,g,Qfor the unknot withp = 4

Q g = 0 g = 1 g = 2 g = 3

−1 2 1 0 00 −6 −5 −1 01 4 4 1 0


Table 14N

,g,Qfor the unknot withp= 4

Q g = 0 g = 1 g = 2 g = 3

−1 4 4 1 00 −10 −15 −7 −11 6 11 6 1

Table 15N

,g,Qfor the unknot with framingp = 4

Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 g = 8 g = 9 g = 10

−3/2 12 26 22 8 1 0 0 0 0 0 0−1/2 −58 −181 −246 −175 −67 −13 −1 0 0 0 0

1/2 86 335 582 550 298 92 15 1 0 0 03/2 −40 −180 −358 −383 −232 −79 −14 −1 0 0 0

Table 16N

,g,Qfor the unknot with framingp = 4

Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 g = 8 g = 9 g = 10

−3/2 46 155 224 167 66 13 1 0 0 0 0−1/2 −206 −915 −1836 −2057 −1377 −561 −136 −18 −1 0 0

1/2 290 1545 3768 5226 4446 2394 817 171 20 1 03/2 −130 −785 −2156 −3336 −3135 −1846 −682 −153 −19 −1 0

Table 17N

,g,Qfor the unknot with framingp= 4

Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 g = 8 g = 9 g = 10

−3/2 40 180 358 383 232 79 14 1 0 0 0−1/2 −170 −965 −2514 −3719 −3367 −1925 −696 −154 −19 −1 0

1/2 230 1535 4746 8446 9374 6748 3196 987 191 21 13/2 −100 −750 −2590 −5110 −6239 −4902 −2514 −834 −172 −20 −1

and the closed Gromov–Witten rational numbers can be deduced from the integerinvariants. In Appendix A, we have theNR,g,Q for representations upto two boxes forthe unknot with arbitrary framing. They will be useful to determineng,m correspondingto Chern–Simons theory on Lens spacesL(2p,1). In the following subsection, we willconsider framed links.


4.3. Framed links

We take Hopf linkH ∗(p1,p2) with linking number�k = −1 and the framing on the twocomponent knots asp1 = p2 = 4. Surgery of such a framed link inS3 will give Lens spaceL(15,4). TheN(R1,R2),g,Q = N(R2,R1),g,Q for this example is shown in Tables 18–26.

When one of the representations is trivial, theN(.,R),g,Q’s will be equal to the invariantsNR,g,Q’s computed for unknot with framingp = 4.

The Gopakumar–Vafa invariants from the expansion of lnZ0[L(15,4)] are

ng,1[L(15,4)

]= 2ng,1[L(4,1)

]+ δg,0,

ng,2[L(15,4)

]= δg,0N( , ),g,−1 + 2ng,2[L(4,1)

],

ng�1,3[L(15,4)

]= N( , ),g,0 − N( , ),g,−1 + 2N( , ),g,−3/2

+ 2ng,3[L(4,1)

],

Table 18N( , ),g,Q

for the framed Hopf linkH ∗(4,4)

Q g = 0

−1 10 −1

Table 19N( , ),g,Q

for H ∗(4,4)

Q g = 0 g = 1 g = 2 g = 3

−3/2 3 1 0 0−1/2 −9 −6 −1 0

1/2 6 5 1 0

Table 20N( , ),g,Q

for H ∗(4,4)

Q g = 0 g = 1 g = 2 g = 3

−3/2 6 5 1 0−1/2 −16 −20 −8 −1

1/2 10 15 7 1

Table 21N( , ),g,Q


Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 g = 8 g = 9 g = 10

−2 23 41 29 9 1 0 0 0 0 0 0−1 −117 −312 −367 −230 −79 −14 −1 0 0 0 0

0 180 606 920 771 376 106 16 1 0 0 01 −86 −335 −582 −550 −298 −92 −15 −1 0 0 0


Table 22N( , ),g,Q


Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 g = 8 g = 9 g = 10

−2 94 271 338 221 78 14 1 0 0 0 0−1 −438 −1697 −3001 −3003 −1820 −680 −153 −19 −1 0 0

0 634 2971 6431 8008 6188 3060 969 190 21 1 01 −290 −1545 −3768 −5226 −4446 −2394 −817 −171 −20 −1 0

Table 23N( , ),g,Q


Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6 g = 7 g = 8 g = 9 g = 10

−2 86 335 582 550 298 92 15 1 0 0 0−1 −376 −1880 −4350 −5776 −4744 −2486 −832 −172 −20 −1 0

0 520 3080 8514 13672 13820 9142 4013 1158 211 22 11 −230 −1535 −4746 −8446 −9374 −6748 −3196 −987 −191 −21 −1

Table 24N( , ),g,Q


Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6

−2 15 17 7 1 0 0 0−1 −60 −83 −45 −11 −1 0 0

0 81 126 75 20 2 0 01 −36 −60 −37 −10 −1 0 0

Table 25N( , ),g,Q


Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6

−2 27 45 30 9 1 0 0−1 −105 −206 −165 −66 −13 −1 0

0 138 301 262 113 24 2 01 −60 −140 −127 −56 −12 −1 0

Table 26N( , ),g,Q


Q g = 0 g = 1 g = 2 g = 3 g = 4 g = 5 g = 6

−2 48 106 99 47 11 1 0−1 −184 −466 −501 −287 −91 −15 −1

0 236 660 767 470 159 28 21 −100 −300 −365 −230 −79 −14 −1


(4.11)

ng�4,4[L(15,4)

]= −N( , ),g,0 − 2N( , ),g,−3/2 + 2N( , ),g,−1/2

− 2N( , ),g,−3/2 + 2N( , ),g,−2 + N( , ),g,−2

+ 2ng,4[L(4,1)

].

These examples suggest that the Chern–Simons partition function on various manifolds canbe given a A-model closed string theory interpretation. From the prediction of Gopakumar–Vafa integer invariants it should be possible to determine the nature of the Calabi–Yaubackground.

5. Summary and conclusions

We have studiedU(N)Chern–Simons gauge theory and framed link invariants inS3, forspecific choice ofU(1) representations placed on the component knots of the link. FromtheseU(N) framed link invariants inS3, we have constructed three-manifold invariantswhich are the same asSU(N) three-manifold invariants. These invariants are proportionalto the Chern–Simons partition functionZ[M] on the corresponding three-manifolds.

In this paper, we have used the results of the topological string duality conjecturerelating Chern–Simons theory onS3 to closed A-model topological string theory onthe resolved conifold, to obtain largeN expansions of the Chern–Simons free-energy(lnZ[M]) on some nontrivial manifolds. The closed string theory expansion resemblesthe A-model topological string theory on a Calabi–Yau space with one Kahler parameter.We have computed the Gopakumar–Vafa integer invariants and the Gromov–Witteninvariants associated with Chern–Simons partition function on three-manifolds like Seifertmanifolds, Lens spaces etc.

We need to understand some subtle issues about the Chern–Simons partition function onany three-manifoldM. As we have pointed out in the introduction, the classical solutionsof the Chern–Simons action are the flat connections onM, and in the weak coupling(largek) limit, the partition function, which may be a sum or integral over the space offlat connections, can be written as

(5.1)Z[M] =∑c

Zc[M] ≡∫µc

Zc[M],

whereZc[M] is obtained from perturbative expansion around a stationary pointA= Ac.The largeN expansion proposed by ’t Hooft requires lnZc[M] to have a closed stringinterpretation whereas we have shown in this paper that lnZ[M] has a closed stringexpansion for many manifolds.

Note that for the case of the three-sphereS3, there is only one stationary point, which isthe trivial connection. ThereforeZ[S3] is also equal to the perturbative expansion aroundthe trivial connection and hence the closed string interpretation is expected from ’t Hooft’sformulation.

In the context of Lens spacesL(p,1), the space of flat connections is a set of points.In Ref. [17],Z[M] has been rewritten as a sum over all flat connections, enabling theextraction of the perturbative partition function around a nontrivial flat connection (Zc[M])


for L(p,1). It has been shown from the matrix model approach thatZc[M] can be givena closed string theoretic interpretation. The results establish the duality between Chern–Simons theory on Lens spacesL(p,1) and closed string theory on aAp−1 singularityfibred overP 1. The Gopakumar–Vafa integer invariants that we have computed for LensspacesL(2p,1) correspond to ln(

∑c Zc[M]). There must be some relation between these

integer invariants and the corresponding invariants onA2p−1 singularity fibred overP 1

at some special values of the Kahler parameters. We hope to decipher such interestingrelations in future.

It appears to be a difficult task to rewriteZ[M] as a sum or integral over flat connectionsto determineZc[M] for other three-manifolds like Seifert-manifolds. The challenge liesin determiningZc[M] and the closed string expansion to precisely state new dualityconjectures between Chern–Simons theory onM and closed string theory. We leave thestudy of these aspects for a future publication.

Acknowledgements

P.R. would like to thank M. Marino and C. Vafa for discussions during the initial stagesof the project, and is grateful to N. Habegger for comments. She would like to thank theAbdus Salam ICTP for providing local hospitality and an excellent academic atmosphereduring her visit under the ICTP Junior Associateship scheme. We would like to thankS. Govindarajan, T.R. Govindarajan, M. Marino, K. Ray and G. Thompson for discussionsand clarifications. P.B. would like to thank CSIR for the grant.

Appendix A. NR,g,Q for the unknot with arbitrary framing p

For an unknot with arbitrary framingp, theNR,g,Q for fundamental representation is

(A.1)N ,0,Q=±1/2 = ∓(−1)p, N ,g =0,Q = 0.

For representations involving two boxes in the Young tableau,N ’s for arbitraryg can bewritten as follows

N ,p−s,−1 ={θ(s − 3)

(s − 3)! (2p− 2s + 2)(2p− 2s + 3) · · ·(2p− s − 2)

+ θ(s − 5)

(s − 5)! (2p− 2s + 2)(2p− 2s + 3) · · ·(2p− s − 4)+ · · ·}(A.2)+ xθ(s − 2),

(A.3)N ,p−s,0 = −δs,2 − θ(s − 2)

(s − 2)! (2p− 2s + 3)(2p− 2s + 4) · · · (2p− s),


N ,p−s,1 ={θ(s − 2)

(s − 2)! (2p− 2s + 2)(2p− 2s + 3) · · · (2p− s − 1)

+ θ(s − 4)

(s − 4)! (2p− 2s + 2)(2p− 2s + 3) · · · (2p− s − 3)+ · · ·}(A.4)+ yθ(s − 1),

N ,p−s,−1 ={θ(s − 2)

(s − 2)! (2p− 2s + 2)(2p− 2s + 3) · · · (2p− s − 1)

+ θ(s − 4)

(s − 4)! (2p− 2s + 2)(2p− 2s + 3) · · · (2p− s − 3)+ · · ·}(A.5)+ yθ(s − 1),

(A.6)N ,p−s,0 = −δs,1 − θ(s − 1)

(s − 1)! (2p− 2s + 3)(2p− 2s + 4) · · ·(2p− s + 1),

N ,p−s,1 ={θ(s − 1)

(s − 1)! (2p− 2s + 2)(2p− 2s + 3) · · · (2p− s)

+ θ(s − 3)

(s − 3)! (2p− 2s + 2)(2p− 2s + 3) · · · (2p− s − 2)+ · · ·}

(A.7)+ xθ(s),

with x = 1 andy = 0 for odds, x = 0 andy = 1 for evens andθ(u) is the usual thetafunction which is equal to one ifu > 0 and zero ifu� 0. The negative framing (−p) BPSintegers are related to the positive framing integers [8]

(A.8)N(p)

(R1,...,Rr ),g,Q= (−1)

∑α �α−1N

(−p)(Rt1,...,R

tr ),g,−Q,

whereRtα ’s are obtained by transposing the rows and columns in the Young tableaurepresentations. These integers are consistent with Marino–Vafa results forg = 0,1,2 [9].

References

[1] R. Gopakumar, C. Vafa, M-theory and topological strings, I, hep-th/9809187.[2] R. Gopakumar, C. Vafa, On the gauge theory/geometry correspondence, hep-th/9811131.[3] R. Gopakumar, C. Vafa, M-theory and topological strings, II, hep-th/9812127.[4] E. Witten, Chern–Simons gauge theory as a string theory, hep-th/9207094.[5] H. Ooguri, C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419, hep-th/9912123.[6] J.M.F. Labastida, M. Marino, Polynomial invariants for torus knots and topological strings, hep-th/0004196.[7] P. Ramadevi, T. Sarkar, On link invariants and topological string amplitudes, Nucl. Phys. B 600 (2001) 487.[8] J.M.F. Labastida, M. Marino, C. Vafa, Knots, links and branes at largeN , JHEP 11 (2000) 007, hep-

th/0010102.[9] M. Marino, C. Vafa, Framed knots at largeN , hep-th/0108064.

[10] M. Aganagic, C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041.[11] M. Aganagic, A. Klemm, C. Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforschr.

A 57 (2002), hep-th/0105045.


[12] S. Govindarajan, T. Jayaraman, T. Sarkar, Disc instantons in linear sigma models, Nucl. Phys. B 646 (2002)498, hep-th/0108234.

[13] W. Lerche, P. Mayr, N. Warner,N = 1 special geometry, mixed Hodge variations and toric geometry, hep-th/0208039.

[14] G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461.[15] W.B.R. Lickorish, 3-manifolds and the temperley Lieb algebra, Math. Ann. 290 (1991) 657;

W.B.R. Lickorish, Three-manifold invariants from combinatorics of Jones polynomial, Pac. J. Math. 149(1991) 337.

[16] R.K. Kaul, P. Ramadevi, Three-manifold invariants from Chern–Simons field theory with arbitrary semi-simple gauge groups, Commun. Math. Phys. 217 (2001) 295, hep-th/0005096.

[17] M. Aganagic, A. Klemm, M. Marino, C. Vafa, Matrix model as a mirror of Chern–Simons theory, hep-th/0211098.

[18] P. Ramadevi, S. Naik, Computation of Lickorish’s three manifold invariants using Chern–Simons theory,Commun. Math. Phys. 209 (2000) 29, hep-th/9901061.

[19] J.M.F. Labastida, M. Marino, A new point of view in the theory of knot and link invariants, J. Knot TheoryRamifications 11 (2002) 173, Math.QA/0104180.


SO(N) reformulated link invariants from topologicalstrings

Pravina Borhade, P. Ramadevi

Department of Physics, Indian Institute of Technology Bombay, Mumbai 400 076, India

Received 7 June 2005; accepted 17 August 2005

Available online 9 September 2005

Abstract

LargeN duality conjecture betweenU(N) Chern–Simons gauge theory onS3 andA-model topologicalstring theory on the resolved conifold was verified at the level of partition function and Wilson loop ob-servables. As a consequence, the conjectured form for the expectation value of the topological operatorsin A-model string theory led to a reformulation of link invariants inU(N) Chern–Simons theory givingnew polynomial invariants whose integer coefficients could be given a topological meaning. We show thatthe A-model topological operator involvingSO(N) holonomy leads to a reformulation of link invariantsin SO(N) Chern–Simons theory. Surprisingly, theSO(N) reformulated invariants also has a similar formwith integer coefficients. The topological meaning of the integer coefficients needs to be explored from theduality conjecture relatingSO(N) Chern–Simons theory toA-model closed string theory on orientifold ofthe resolved conifold background. 2005 Elsevier B.V. All rights reserved.

1. Introduction

Within the last one decade we have seen interesting developments in the open string and closedstring dualities. One such open-closed string duality conjecture relatesA-model open topologicalstring theory on the deformed conifold, equivalent to Chern–Simons gauge theory onS3 [1], tothe closed string theory on a resolved conifold.

Gopakumar–Vafa[2–4] showed that the free-energy expansion ofU(N) Chern–Simons fieldtheory onS3 at largeN resemblesA-model topological string theory amplitudes on the resolvedconifold. The conjecture was further tested at the level of observables in Chern–Simons theory.

E-mail addresses:[email protected](P. Borhade),[email protected](P. Ramadevi).

0550-3213/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2005.08.027

mailto:[email protected]


http://dx.doi.org/10.1016/j.nuclphysb.2005.08.027

472 P. Borhade, P. Ramadevi / Nuclear Physics B 727 [PM] (2005) 471–498

Ooguri–Vafa[5] considered the expectation value of a topological operator corresponding to asimple circle (called unknot) in submanifoldS3 of the deformed conifold and showed its formin the resolved conifold background. The results led to a new conjecture (usually referred to asOoguri–Vafa conjecture) on the form for the expectation value of the topological operator forany knot or link inS3.

Using group theory, Labastida–Marino[6] showed that the expectation value of the topolog-ical operators can be rewritten in terms of link invariants inU(N) Chern–Simons field theoryon S3. This enabled verification of Ooguri–Vafa conjecture for many non-trivial knots[6–9].Conversely, the Ooguri–Vafa conjecture led to a reformulation of Chern–Simons field theoryinvariants for links giving new polynomial invariants. The integer coefficients of these new poly-nomial invariants have topological meaning accounting for BPS states in the string theory. Thechallenge still remains in obtaining such integers within topological string theory.

Similar duality conjectures have been attempted between Chern–Simons gauge theories onthree-manifolds other thanS3 and closed string theories. In Ref.[10], U(N) Chern–Simons free-energy expansion at largeN for many three-manifolds were derived and the expansion resembledpartition function of a closed string theory on a Calabi–Yau background with one Kähler para-meter. Unfortunately, the Chern–Simons free-energy expansion for other three-manifolds are notequivalent to the ’t Hooft largeN perturbative expansion around a classical solution[11]. Hencewe need to extract the perturbative expansion around a classical solution from the free-energy toobtain new duality conjectures.

For orbifolds ofS3, which gives Lens spaceL[p,1] ≡ S3/Zp, it is believed that the Chern–Simons theory is dual to theA-model closed string theory onAp−1 fibred overP 1 Calabi–Yau background. It was Marino[12] who showed that the perturbative Chern–Simons theoryon Lens spaceL[p,1] can be given a matrix model description. Also, Hermitian matrix modeldescription ofB-model topological strings[13] was shown to be equivalent to Marino’s matrixmodel using mirror symmetry[14]. It is still a challenging open problem to look for dual closedstring description corresponding toU(N) Chern–Simons theory on other three-manifolds.

The extension of these duality conjectures for other gauge groups likeSO(N) andSp(N) havealso been studied. In particular, the free-energy expansion of the Chern–Simons theory based onSO/Spgauge group was shown to be dual toA-model closed string theory on a orientifold of theresolved conifold background[15]. Further, using the topological vertex as a tool, Bouchard etal. [16,17]have determined unoriented closed string amplitude and unoriented open topologicalstring amplitudes for a few orientifold toric geometry with or without D-branes.

It will be interesting to generalise Ooguri–Vafa conjecture by looking at the topological op-erator involvingSO/Spholonomy instead of theU(N) holonomy. In this paper, we obtain newreformulated polynomial invariants in terms of the framed link polynomials inSO(N) Chern–Simons theory. Similar to theU(N) result, the coefficients are indeed integers and the topologicalmeaning in terms of the BPS invariants in string theory needs to be explored. Further, the refor-mulated invariant for knots in standard framing obeys the conjecture of Bouchard–Florea–Marino[17] giving the integer coefficients corresponding to cross-capc = 1 unoriented open-string am-plitude. We generalise the conjecture for anyr-component framed links and have verified for fewexamples of framed knots and two-component framed links.

The organisation of the paper is as follows. In Section2, we present framed link invariants inSO(N) Chern–Simons theory. In Section3, first we recapitulate the topological operator carryingU(N) holonomy and then elaborate its generalization toSO(N) holonomy. Section4 containssome explicit results of the reformulated polynomial invariants. We present the integer coeffi-cients in the cross-capc = 1 unoriented string amplitudes for few framed knots and links in

P. Borhade, P. Ramadevi / Nuclear Physics B 727 [PM] (2005) 471–498 473

Section5. In the concluding Section6, we summarize the results obtained and pose open ques-tions for future research. InAppendix A, we presentSO(N) polynomials for few framed knotsand framed links for some representations. InAppendix B, the reformulated polynomial invari-ants for few non-trivial framed knots and framed links are presented.

2. SO(N) Chern–Simons gauge theory and framed link invariants

Chern–Simons gauge theory onS3 based on the gauge groupSO(N) is described by thefollowing action:

(2.1)S = k

4π

∫S3

Tr

(A ∧ dA + 2

3A ∧ A ∧ A

),

whereA is a gauge connection for gauge groupSO(N) and k is the coupling constant. Theobservables in this theory are Wilson loop operators

(2.2)WR1,R2,...,Rr [L] =r∏

i=1

TrRiU [Ki],

whereU [Ki] = P [exp∮Ki

A] denotes the holonomy of theSO(N) gauge fieldA around thecomponent knotKi of a r-component linkL carrying representationRi . The expectation valueof these Wilson loop operators are theSO(N) link invariants

(2.3)VΛR1,ΛR2,...,ΛRr[L](q,λ) = ⟨

WR1,R2,...Rr [L]⟩(q,λ) =∫ [DA]eiSWR1,R2,...,Rr [L]∫ [DA]eiS

,

whereΛRi’s denote the highest weights of the representationRi ’s. TheSO(N) link invariants

are polynomials in two variables

(2.4)q = exp

(2πi

k + N − 2

), λ = qN−1

involving the coupling constantk and the rank of the gauge group. These link invariants can becomputed using the following two inputs: (i) any link can be drawn as a closure or plat of braids,(ii) the connection between Chern–Simons theory and the Wess–Zumino conformal field theory.

The invariant for the unknot is equal to the quantum dimension of the representationR livingon the unknot

(2.5)VΛR[U ](q,λ) = dimq R,

where the quantum dimension of the representationR with highest weightΛR is given by

(2.6)dimq R = Πα>0[α · (ρ + ΛR)]

[α · ρ] ,

whereα’s are the positive roots andρ is the Weyl vector equal to the sum of the fundamentalweights of the groupSO(N). The square bracket refers to the quantum number defined by

(2.7)[x] = (qx/2 − q−x/2)

(q1/2 − q−1/2).


We shall now present the polynomials for various framed knots and links. For the unknotU

with an arbitrary framingp, carrying a representationR of SO(N), the polynomial is

(2.8)VΛR

[0(p)

](q,λ) = (−1)�pq(pCR)VΛR

[U ](q,λ),

where� refers to the total number of boxes in the Young tableau of the representationR and thequadratic CasimirCR = (ΛR+2ρ)·ΛR

2 in terms of Young tableau is given by

CR = 1

2

((N − 1)� + � +

∑i

(l2i − 2ili

)).

Hereli denotes the number of boxes in theith row of the Young tableau of the representationR.Eqs.(A.1)–(A.11) in Appendix Acontain explicitp-framed unknot polynomials for few repre-sentations.

Now, we can write theSO(N) framed knot invariants for torus knots of the type(2,2m + 1)

with framing[p − (2m + 1)] as follows:

(2.9)VΛR[K](q,λ) = (−1)�[p−(2m+1)]qpCR

∑Rs∈R⊗R

dimq Rs(−1)εs(qCR−CRs /2)2m+1

,

whereεs = ±1 depending upon whether the representationRs appears symmetrically or antisym-metrically with respect to the tensor productR ⊗R in theSO(N)k Wess–Zumino–Witten model.Explicit polynomial expression forp framed trefoil for some representations are presented inAppendix A.

Similarly, SO(N) invariants for framed torus links of the type(2,2m) can also be written. Forexample, theSO(N) invariant for a Hopf link with linking number−1 and framing numbersp1

andp2 on the component knots carrying representationsR1 andR2 will be

VΛR1,ΛR2

[H ∗(p1,p2)

](q,λ)

(2.10)= (−1)�1p1+�2p2qp1CR1+p2CR2∑

Rs∈R1⊗R2

dimq RsqCR1+CR2−CRs .

We have presented the explicit framed Hopf link polynomials for some representations inAp-pendix A. Using the framed torus knot/link invariants, we can write theSO(N) invariants forconnected sums. For example, knotK = K1#K2, whereK1 andK2 are framed torus knots,

(2.11)VΛR1[K = K1#K2](q,λ) = 1

VΛR1[U ](q,λ)

(VΛR1

[K1](q,λ)VΛR1[K2](q,λ)

).

We can also consider a linkL obtained as a connected sum of a torus knotK1 and a torus linkL1. The link invariant will be

(2.12)

VΛR1,R2[L = K1#L1](q,λ) = 1

VΛR1[U ](q,λ)

(VΛR1

[K1](q,λ)VΛR1,ΛR2[L1](q,λ)

).

In the following section, we will see the reformulation ofSO(N) invariants giving new polyno-mial invariants.


3. Reformulated link invariants

We will briefly review the new polynomial invariants obtained as a reformulation of link in-variants inU(N) Chern–Simons theory. Then, we address the modified group theoretic equationsfor theSO(N) group and show that they also give a similar reformulated invariants.

3.1. U(N) reformulated link invariants

Ooguri and Vafa showed that the Wilson loop operators in Chern–Simons theory correspondto certain observables in the topological string theory giving another piece of evidence forGopakumar–Vafa duality conjecture. The operators in the open topological string theory whichcontains information about links is given by[5]

(3.1)Z({Uα}, {Vα})= exp

[r∑

α=1

∞∑d=1

1

dTrUd

α TrV dα

]whereUα is the holonomy of the gauge connectionA around the component knotKα carryingthe fundamental representation in theU(N) Chern–Simons theory onS3, andVα is the holonomyof a gauge fieldA around the same component knot carrying the fundamental representation intheU(M) Chern–Simons theory on a Lagrangian three-cycle which intersectsS3 along the curveKα . The above operator can be equivalently represented as

(3.2)Z({Uα}, {Vα})= 1+

∑{�k(α)}

r∏α=1

1

z�k(α)

γ�k(α)(Uα)γ�k(α) (Vα),

where

(3.3)z�k(α) =∏j

k(α)j !jk

(α)j , γ�k(α) (Uα) =

∞∏j=1

(TrUj

α

)k(α)j .

Here �k(α) = (k(α)1 , k

(α)2 , . . .) with |�k(α)| =∑

j k(α)j and the sum is over all the vectors�k(α) such

that∑r

α=1 |�k(α)| > 0. Using the following group theoretic Frobenius equations,

(3.4)γk1(U1) · · ·γkr (Ur) =∑

R1,...,Rr

r∏α=1

χRα

(C(�k(α)

))TrR1(U1) · · ·TrRr (Ur),

(3.5)∑

�k

1

z�kχR1

(C(�k)

)χR2

(C(�k)

)= δR1R2,

whereχRα (C(�k(α)))’s are characters of the symmetry groupS�α with �α =∑j jk

(α)j andC(�k(α))

are the conjugacy classes associated to�k(α)’s (denotingk(α)j cycles of lengthj ), the operator can

be shown to be

(3.6)Z({Uα}, {Vα})=

∑{Rα}

r∏α=1


Ooguri and Vafa have conjectured a specific form for the vacuum expectation value (vev) of thetopological operators(3.1) for knots [5] invoking the largeN topological string duality. This


result was further refined for links[8] which is generalisable for framed links[9] as follows

(3.7)⟨Z({Uα}, {Vα})⟩

A= exp

[ ∞∑d=1

∑{Rα}

1

df(R1,...,Rr )

(qd,λd

) r∏α=1

TrRα V dα

],

(3.8)f(R1,R2,...,Rr )(q, λ) =∑Q,s

1

(q1/2 − q−1/2)N(R1,...,Rr ),Q,sq

sλQ,

where the suffixA on the vev implies that the expectation value is obtained by integrating theU(N) gauge fieldsA’s on S3. Further, for framed linksN(R1,...,Rr ),Q,s are integers. In fact,fR1,R2,...,Rr (q, λ) are theU(N) reformulated polynomial invariants involvingU(N) Chern–Simons link invariants.

The general formula for the reformulated polynomial invariantf (3.8) in terms ofU(N)

framed link invariantsV {U(N)}ΛR1j

,ΛR2j,...,ΛRrj

[L,S3](qd, λd) [10] can be written as[18]

fR1,R2,...Rr (q, λ)

=∞∑

d,m=1

(−1)m−1µ(d)

dm


r∏α=1

χRα

(C

((m∑

j=1

�k(αj)

)d

))m∏

j=1

|C(�k(αj))|�αj !

(3.9)× χRαj

(C(�k(αj)

))V

{U(N)}ΛR1j

,ΛR2j,...,ΛRrj

[L,S3](qd,λd

),

whereµ(d) is the Moebius function defined as follows: ifd has a prime decomposition ({pi}),d =∏a

i=1 pmi

i , thenµ(d) = 0 if any of themi is greater than one. If allmi = 1, thenµ(d) =(−1)a . The second sum in the above equation runs over all vectors�k(αj), with α = 1, . . . , r andj = 1, . . . ,m, such that

∑rα=1 |�k(αj)| > 0 for any j and over representationsRαj . Further�kd

is defined as follows:(�kd)di = ki and has zero entries for the other components. Therefore, if�k = (k1, k2, . . .), then

(3.10)�kd = (0, . . . ,0, k1,0, . . . ,0, k2,0, . . .),

wherek1 is in thed th entry,k2 in the 2d th entry, and so on. Hence, one can directly evaluatef

from U(N) framed link invariants and verify the conjecture(3.8). Further refinement of Eq.(3.9)revealing the BPS structure has been presented in[8]

(3.11)fR1,R2,...,Rr (q, λ) =∑

R′1,...,R

′r

MR1,...,Rr ;R′1,...,R

′rf(R′

1,...,R′r )(q, λ),

where

(3.12)MR1,...,Rr ;R′1,...,R

′r=

∑R′′

1,...R′′r

r∏α=1

CRαR′αR′′

αSR′′

α(q),

(3.13)f(R′1,...,R

′r )(q, λ) = (

q−1/2 − q1/2)r−2 ∑g�0,Q

N(R′1,...,R

′r ),g,Q

(q−1/2 − q1/2)2g

λQ.

In Eq. (3.12), Rα,R′α,R′′

α are representations of the symmetric groupS�α which can be la-belled by a Young tableau with a total of�α boxes andCRR′R′′ are the Clebsch–Gordan co-efficients of the symmetric group.SR(q) is non-zero only for hook representations. For suchhook representation having� − d boxes in the first row of the Young tableau with total� boxes,SR(q) = (−1)dq−(�−1)/2+d .


3.2. SO(N) reformulated invariants

As a problem within Chern–Simons field theory, we could take the same operator(3.1) tocarry SO(N) holonomy instead ofU(N) holonomy. We shall denote the topological operatorinvolving SO(N) holonomy as

(3.14)Z({Uα}, {Vα})= exp

[r∑

α=1

∞∑d=1

1

dTr U d

α Tr V dα

],

whereUα is the holonomy of the gauge connectionA around the component knotKα carryingthe defining representation in theSO(N) Chern–Simons theory onS3, andVα is the holonomyof a gauge fieldA around the same component knot carrying the defining representation in theSO(M) Chern–Simons theory on a Lagrangian three-cycleC which intersectsS3 along the curveKα . In the context of the duality of theSO Chern–Simons theory to the closed string on theorientifold of the resolved conifold[15], the gauge group of the Chern–Simons theory onS3

alone has to beSO. Even if we chooseSO gauge group for Chern–Simons theory onS3 andSOChern–Simons theory on Lagrangian cycleC, the results relevant to open topological stringamplitude[17] in the orientifold background will be unaltered.

Similar to Eq.(3.2), the above operator can be equivalently represented as

(3.15)Z({Uα}, {Vα})= 1+

∑{�k(α)}

r∏α=1

1

z�k(α)

γ�k(α)(Uα)γ�k(α) (Vα)

with the usual definitions forz�k(α) andγ�k(α) (Uα) as given in Eq.(3.3).Now, we need to modify the Frobenius equation which is one of the main results of the paper.

The orthogonality relation Eq.(3.5)remains the same for theSOcase. Eq.(3.4)will be modifiedas follows:

(3.16)γk1(U1) · · ·γkr (Ur) =∑

R1,...,Rr

r∏α=1

χRα

(C(�k(α)

))TrR1(U1) · · · TrRr (Ur),

whereχRα (C(�k(α)))’s are again characters of the symmetry groupS�α with �α =∑j jk

(α)j and

C(�k(α)) are the conjugacy classes associated to�k(α)’s (denotingk(α)j cycles of lengthj ). Notice

that we have put a ‘hat’ in the trace function in the above equation. We shall explain the meaningof the ‘hat’ by presentingTrR(U) for few SO(N) representations (we denote representationR

by the highest weightΛR for convenience)

Trλ(1)U = Trλ(1) U,

Tr2λ(1)U = Tr2λ(1) U + 1,


Tr3λ(1)U = Tr3λ(1) U + Trλ(1) U,

Trλ(1)+λ(2)U = Trλ(1)+λ(2) U + Trλ(1) U,


Tr4λ(1)U = Tr4λ(1) U + Tr2λ(1) U + 1,

Tr2λ(1)+λ(2)U = Tr2λ(1)+λ(2) U + Tr2λ(1) U + Trλ(2) U,


Tr2λ(2)U = Tr2λ(2) U + Tr2λ(1) U + 1,

Trλ(1)+λ(3)U = Trλ(1)+λ(3) U + Trλ(2) U,

(3.17)Trλ(4)U = Trλ(4) U.

In principle, Tr can be derived for arbitrarySO(N) representation with highest weightΛR =∑i=1 n

(R)i λ(i). In the next section, we will use the above set ofTr (3.17)for obtaining explicit

results on topological open-string amplitudes.Using Eqs.(3.16), (3.5), it is not difficult to see that theSO topological operator(3.15) is

equivalent to

(3.18)Z({Uα}, {Vα})=

∑{Rα}

r∏α=1


Similar to Ooguri–Vafa conjecture, we propose the following conjecture for the operator(3.14)involving SOholonomy:

Conjecture 1.

(3.19)eF({Vα}) = ⟨Z({Uα}, {Vα})⟩

A= exp

[ ∞∑d=1

∑{Rα}

1

dg(R1,...,Rr )

(qd,λd

) r∏α=1

TrRα V dα

],

(3.20)g(R1,R2,...,Rr )(q, λ) =∑Q,s

1

(q1/2 − q−1/2)N(R1,...,Rr ),Q,sq

sλQ,

where the suffixA on the vev implies that the expectation value is obtained by integrating theSO(N) gauge fieldsA’s onS3 andN(R1,...,Rr ),Q,s in Eq. (3.20)are integers.

We have introducedF({Vα}) which we call as open-string partition function. Incidentally,F({Vα}) is a sum of oriented string partition function(untwisted sector) and unoriented stringpartition function (twisted sector) as presented in Ref.[17].

The functiongR1,R2,...,Rr (q, λ) are theSO(N) reformulated polynomial invariants involvingframed link invariants inSO(N) Chern–Simons theory. It is easy to see that Eq.(3.9) can beaccordingly modified forSOgroup involving the expectation value ofTrRU (3.17)as follows:

gR1,R2,...,Rr (q, λ)

=∞∑

d,m=1

(−1)m−1µ(d)

dm


r∏α=1

χRα

(C

((m∑

j=1

�k(αj)

)d

))m∏

j=1

|C(�k(αj))|�αj !

(3.21)× χRαj

(C(�k(αj)

))⟨ r∏α=1

TrRαjUα[Kα]

⟩(qd,λd

),

where the definitions ofµ(d), �k(αj) and�kd are same as defined in the previous subsection. Sub-stituting theTr (3.17) and rewriting in terms ofSO(N) framed link invariants(2.3), we haveexplicitly verified that theSO(N) reformulated invariant obeys the conjectured Eq.(3.20) formany framed knots and framed links. This is one of the non-trivial results of the paper which wepresent in the next section and inAppendix B.


Using Eq. (3.12), we can rewrite theSO(N) reformulated polynomial invariantsgR1,R2,...,Rr (q, λ) as

(3.22)gR1,R2,...,Rr (q, λ) =∑

R′1,...,R

′r

MR1,...,Rr ;R′1,...,R

′rg(R′

1,...,R′r )(q, λ).

Unfortunately,gR1,...,Rr (q, λ) does not have a BPS structure like the one given in Eq.(3.13)forU(N) holonomy. This has also been extensively studied in the works of Bouchard et al.[17]where they conjecture an equation forgR(q,λ) corresponding to knots in standard framing asfollows:

(3.23)1

2

(gR

(q,λ1/2)− (−1)�(R)gR

(q,−λ1/2))=

∑g,β

Nc=1R,g,β

(q1/2 − q−1/2)2g

λβ.

In this equation,Nc=1R,g,β are BPS invariants corresponding to unoriented open string amplitudes

with one cross-cap. The above conjecture can be generalised for arbitraryr-component framedlinks [L,p] wherep = (p1,p2, . . . , pr) denotes the framing numberspi ’s on the componentknotsKi ’s. For suchr-component framed links[L,p], we propose the following conjecture:

Conjecture 2.

1

2

(gR1,R2,...,Rr

(q,λ1/2)− (−1)

∑rα=1 �(Rα)(pα+1)gR1,R2,...,Rr

(q,−λ1/2))

(3.24)=∑g,β

Nc=1R1,R2,...,Rr ,g,β

(q1/2 − q−1/2)2g+r−1

λβ.

We have verified the above conjecture for many framed knots and framed two component links.In Section5, we have presentedNc=1

R1,R2,...,Rr ,g,β for some framed knots and framed two compo-nent links.

4. Explicit computation of SO(N) reformulated invariants gR1,R2,...,Rr (q,λ)

In this section we compute the functionsgR1,...,Rr (q, λ) for various non-trivial framed knotsand links and show that they obey the conjectured form(3.20). We shall denote the representa-tionsRi ’s in gR1,...,Rr (q, λ) by their highest weightsΛRi

’s.

• For unknot in standard framing, the reformulated invariant isnon-zero only for the definingrepresentation:

(4.1)Vλ(1)[U ](q,λ) = gλ(1) (q, λ) = 1

q − 1

[q − 1+ q1/2(−1+ λ)λ−1/2].

This simplifies the form for open-string partition functionF({Vα}) in Eq.(3.19)as follows:

(4.2)F(V ) =∑d

1

d

(1+ λd/2 − λ−d/2

qd/2 − q−d/2

)Tr V d .

• For unknot with arbitrary framingp

(4.3)gλ(1) (q, λ) = (−1)pλp/2(

1+ q1/2(−1+ λ)

(−1+ q)λ1/2

),


g2λ(1) (q, λ) = 1

2(−1+ q)2(1+ q)

[2(−1+ q)2(1+ q)

+ λ−1+p(2qp(−1+ λ)

(−q + q1/2(−1+ q2)λ1/2 + q2λ)

+ (−1)p(−((−1)p(1+ q)

(q1/2 + λ1/2)2(−1+ q1/2λ1/2)2)

(4.4)− (−1+ q)(q + λ)(−1+ qλ)))]

,

gλ(2) (q, λ) = 1

2(−1+ q)2(1+ q)

[λ−1+p

(2(q3/2 + λ1/2)(−1+ q1/2λ1/2)q1/2−p(−1+ λ)

− (1+ q)(q1/2 + λ1/2)2(−1+ q1/2λ1/2)2

(4.5)+ (−1)p(−1+ q)(q + λ)(−1+ qλ))]

,

g3λ(1) (q, λ) = 1

(−1+ q)3(1+ q)(1+ q + q2)

[(−1)p

(−1+ qp)(

q1/2 + λ1/2)× (−1+ q1/2λ1/2)(−1+ λ)q1/2λ3(−1+p)/2

× (λ1/2 − q3/2(−1+ q(−1+ λ) + q5/2λ1/2 + λ

)+ qp

(−q1/2 + q(−1+ q2)λ1/2 + q7/2λ

)(4.6)+ q2p

(−q1/2 + q(−1+ q2)λ1/2 + q7/2λ

))],

gλ(1)+λ(2) (q, λ) = −1

(−1+ q)3(1+ q)

[(−1)pq1/2−p

(−1+ qp)(

q1/2 + λ1/2)× (−1+ q1/2λ1/2)(−1+ λ)λ3(−1+p)/2(qp

(q1/2 + λ1/2)

(4.7)× (−1+ q3/2λ1/2)+ q3/2 + λ1/2 − q2λ1/2 − q1/2λ)]

,

gλ(3) (q, λ) = 1

(−1+ q)3(1+ q)(1+ q + q2)

[(−1)pq1/2−3p

(−1+ qp)

× (q1/2 + λ1/2)(−1+ q1/2λ1/2)(−1+ λ

)λ3(−1+p)/2

× (q7/2 − q

(−1+ q2)λ1/2 − q1/2λ + qp(q7/2 − q

(−1+ q2)λ1/2 − q1/2λ)

(4.8)+ q2p(−(q3/2(1+ q)

)+ (−1+ q4)λ1/2 + q3/2(1+ q)λ))]

.

Substituting values forp, the above equations reduce to the conjectured form(3.20).(1) For unknot with framingp = 1, we get

gλ(1) (q, λ) = −1

q − 1

[(q − 1)λ1/2 + q1/2(−1+ λ)

],

g2λ(1) (q, λ) = 1

q − 1

[(−1+ q1/2λ1/2)(1+ q(−1+ λ) + q1/2λ3/2)],gλ(2) (q, λ) = −1

q − 1

[(−q−1/2λ1/2 + λ)(−1+ q + q1/2λ1/2 + λ

)].


(2) For unknot with framing two

gλ(1) (q, λ) = 1

q − 1

[(q − 1)λ + q1/2(−1+ λ)λ1/2],

g2λ(1) (q, λ) = 1

q − 1

[(−1+ λ)

{1− q + λ − 2qλ + q1/2(−1+ q2)λ3/2 + q2λ2}],

(4.9)gλ(2) (q, λ) = −1

q − 1

[q−3/2(q3/2 + λ1/2)(−1+ q1/2λ1/2)(−1+ λ)λ

].

• We have presented the reformulated invariants for few framed knots and two componentlinks in Appendix B.

5. Nc=1(R1,...,Rr ),g,Q computation

We shall now compute the integer coefficients(3.24)corresponding to cross-capc = 1 un-oriented open string amplitude obtained fromSO(N) reformulated invariants for various framedknots and framed links.

5.1. Framed knots

(1) For unknot with zero framing, the only non zero coefficient isNc=1λ(1),0,0

= 1.(2) For unknot with framingp = 1

(5.1)Nc=1λ(1),0,1/2 = −1.

β = 1/2 3/2

g = 0 1 −1

Nc=1λ(2),g,β

β = 3/2 5/2

g = 0 1 −1

Nc=1λ(1)+λ(2),g,β

β = 1/2 3/2 5/2

g = 0 −1 4 −3

1 0 1 −1

Nc=1λ(3),g,β

(3) For unknot with framingp = 2:

(5.2)Nc=1λ(1),0,1 = 1.

β = 3/2 5/2

g = 0 1 −1

Nc=12λ(1),0,3/2


β = 3/2 5/2

g = 0 3 −3

1 1 −1

Nc=1λ(2),g,β

β = 2 3 4

g = 0 1 −4 3

1 0 −1 1

Nc=13λ(1),g,β

β = 2 3 4

g = 0 9 −28 19

1 6 −27 212 1 −9 83 0 −1 1

Nc=1λ(1)+λ(2),g,β

β = 2 3 4

g = 0 13 −36 23

1 16 −57 412 7 −36 293 1 −10 94 0 −1 1

Nc=1λ(3),g,β

(4) For trefoil knot in standard framing, the results are agreeing with the tables given inRef. [17].

(5) For trefoil knot with framingp = 1

β = 3/2 5/2 7/2

g = 0 −3 3 −1

1 −1 1 0

Nc=1λ(1),g,β

β = 5/2 7/2 9/2 11/2 13/2

g = 0 16 −69 111 −79 21

1 20 −146 307 −251 702 8 −128 366 −330 843 1 −56 230 −220 454 0 −12 79 −78 115 0 −1 14 −14 16 0 0 1 −1 0

Nc=12λ(1),g,β

β = 5/2 7/2 9/2 11/2 13/2

g = 0 30 −114 167 −111 28

1 55 −311 587 −457 1262 36 −367 912 −791 2103 10 −230 770 −715 1654 1 −79 376 −364 665 0 −14 106 −105 136 0 −1 16 −16 17 0 0 1 −1 0

Nc=1λ(2),g,β


(6) For trefoil knot with framingp = 2

β = 2 3 4

g = 0 3 −3 1

1 1 −1 0

Nc=1λ(1),g,β

β = 7/2 9/2 11/2 13/2 15/2

g = 0 30 −114 167 −111 28

1 55 −311 587 −457 1262 36 −367 912 −791 2103 10 −230 770 −715 1654 1 −79 376 −364 665 0 −14 106 −105 136 0 −1 16 −16 17 0 0 1 −1 0

Nc=12λ(1),g,β

β = 7/2 9/2 11/2 13/2 15/2

g = 0 50 −174 237 −149 36

1 125 −601 1042 −776 2102 120 −919 2046 −1709 4623 55 −771 2222 −2001 4954 12 −376 1443 −1365 2865 1 −106 574 −560 916 0 −16 137 −136 157 0 −1 18 −18 18 0 0 1 −1 0

Nc=1λ(2),g,β

(7) For connected sum trefoil # trefoil with zero framing

β = 2 3 4 5

g = 0 8 −14 9 −2

1 6 −11 6 −12 1 −2 1 0

Nc=1λ(1),g,β


β = 7/2 9/2 11/2 13/2 15/2 17/2 19/2 21/2

g = 0 143 −831 1950 −2366 1561 −525 66 2

1 404 −3144 8854 −11819 7544 −1488 −596 2452 464 −5419 19211 −28097 14046 6348 −9194 26413 277 −5379 25184 −40255 6296 44160 −41756 114734 90 −3292 21666 −38551 −18588 110890 −98450 262355 15 −1256 12654 −26241 −38613 159091 −141400 357506 1 −290 5048 −13093 −36589 147270 −133378 310317 0 −37 1352 −4787 −21053 92681 −85919 177638 0 −2 232 −1243 −7860 40544 −38455 67849 0 0 23 −215 −1917 12353 −11954 1710

10 0 0 1 −22 −295 2574 −2531 27311 0 0 0 −1 −26 350 −348 2512 0 0 0 0 −1 28 −28 113 0 0 0 0 0 1 −1 0

Nc=12λ(1),g,β

β = 7/2 9/2 11/2 13/2 15/2 17/2 19/2 21/2

g = 0 227 −1237 2756 −3206 2045 −671 84 2

1 801 −5621 14872 −19187 12269 −2861 −564 2912 1190 −11771 38341 −54346 29773 5595 −12485 36973 955 −14403 59796 −92245 27489 67819 −68353 189424 444 −11132 61614−104212 −18925 211571 −190697 513375 119 −5578 43750 −83517 −79482 364896 −323843 836556 17 −1803 21761 −49317 −100043 404876 −363462 879717 1 −362 7561 −21758 −73516 307780 −281842 621368 0 −41 1795 −7097 −35330 165164 −154547 300569 0 −2 277 −1653 −11446 63250 −60402 9976

10 0 0 25 −258 −2483 17202 −16719 223311 0 0 1 −24 −346 3248 −3201 32212 0 0 0 −1 −28 405 −403 2713 0 0 0 0 −1 30 −30 114 0 0 0 0 0 1 −1 0

Nc=1λ(2),g,β

5.2. Framed links

(1) Hopf linkWe take Hopf linkH(p1,p2) with linking number−1 and framing on the two component knotsasp1 andp2. The integersNc=1

(R1,R2),g,β for various combinations ofp1 andp2 are tabulatedbelow.p1 = 0= p2

(5.3)Nc=1(λ(1),λ(1)),0,1/2 = 1.


β = −1 0

g = 0 −1 1

Nc=1(2λ(1),λ(1)),g,β

β = 0 1

g = 0 1 −1

Nc=1(λ(2),λ(1)),g,β

β = −3/2 −1/2

g = 0 −1 1

Nc=1(3λ(1),λ(1)),g,β

β = −1/2 1/2

g = 0 1 −1

Nc=1(λ(1)+λ(2),λ(1)),g,β

β = 1/2 3/2

g = 0 −1 1

Nc=1(λ(3),λ(1)),g,β

p1 = 1 = p2

β = 1/2 3/2

g = 0 −1 1

Nc=1(λ(1),λ(1)),g,β

β = 3/2 5/2

g = 0 −1 1

Nc=1(2λ(1),λ(1)),g,β

β = 1/2 3/2 5/2

g = 0 1 −5 4

1 0 −1 1

Nc=1(λ(2),λ(1)),g,β

β = 5/2 7/2

g = 0 −1 1

Nc=1(3λ(1),λ(1)),g,β

β = 3/2 5/2 7/2

g = 0 6 −19 13

1 1 −8 72 0 −1 1

Nc=1(λ(1)+λ(2),λ(1)),g,β

β = 1/2 3/2 5/2 7/2

g = 0 −1 15 −36 22

1 0 7 −29 222 0 1 −9 83 0 0 −1 1

Nc=1(λ(3),λ(1)),g,β

p1 = 2 = p2

β = 3/2 5/2

g = 0 −1 1

Nc=1(λ(1),λ(1)),g,β

β = 2 3 4

g = 0 −1 5 −4

1 0 1 −1

Nc=1(2λ(1),λ(1)),g,β

β = 2 3 4

g = 0 −4 13 −9

1 −1 7 −62 0 1 −1

Nc=1(λ(2),λ(1)),g,β


β = 5/2 7/2 9/2 11/2

g = 0 −1 15 −36 22

1 0 7 −29 222 0 1 −9 83 0 0 −1 1

Nc=1(3λ(1),λ(1)),g,β

β = 5/2 7/2 9/2 11/2

g = 0 −13 106 −204 111

1 −7 118 −319 2082 −1 55 −219 1653 0 12 −78 664 0 1 −14 135 0 0 −1 1

Nc=1(λ(1)+λ(2),λ(1)),g,β

β = 5/2 7/2 9/2 11/2

g = 0 −22 136 −231 117

1 −22 231 −521 3122 −8 173 −532 3673 −1 67 −296 2304 0 13 −92 795 0 1 −15 146 0 0 −1 1

Nc=1(λ(3),λ(1)),g,β

p1 = 2,p2 = 3

β = 2 3

g = 0 1 −1

Nc=1(λ(1),λ(1)),g,β

β = 5/2 7/2 9/2

g = 0 1 −5 4

1 0 −1 1

Nc=1(2λ(1),λ(1)),g,β


β = 5/2 7/2 9/2

g = 0 4 −13 9

1 1 −7 62 0 −1 1

Nc=1(λ(2),λ(1)),g,β

β = 3 4 5 6

g = 0 1 −15 36 −22

1 0 −7 29 −222 0 −1 9 −83 0 0 1 −1

Nc=1(3λ(1),λ(1)),g,β

β = 3 4 5 6

g = 0 13 −106 204 −111

1 7 −118 319 −2082 1 −55 219 −1653 0 −12 78 −664 0 −1 14 −135 0 0 1 −1

Nc=1(λ(1)+λ(2),λ(1)),g,β

β = 3 4 5 6

g = 0 22 −136 231 −117

1 22 −231 521 −3122 8 −173 532 −3673 1 −67 296 −2304 0 −13 92 −795 0 −1 15 −146 0 0 1 −1

Nc=1(λ(3),λ(1)),g,β

(2) Trefoil # Hopf link

We consider the link whose one component is trefoil and other is Hopf link. We tabulate

Nc=1R1,R2,g,β for the case where both the components carry no framing.

β = 1/2 3/2 5/2 7/2

g = 0 −3 6 −4 1

1 −1 2 −1 0

Nc=1(λ(1),λ(1)),g,β


β = 1 2 3 4 5 6

g = 0 −14 87 −218 266 −157 36

1 −11 113 −415 635 −427 1052 −2 55 −330 650 −485 1123 0 12 −132 351 −285 544 0 1 −26 104 −91 125 0 0 −2 16 −15 16 0 0 0 1 −1 0

Nc=1(2λ(1),λ(1)),g,β

β = 1 2 3 4 5 6

g = 0 −24 154 −363 410 −226 49

1 −26 282 −910 1271 −813 1962 −9 209 −989 1728 −1233 2943 −1 77 −572 1275 −989 2104 0 14 −182 545 −454 775 0 1 −30 135 −120 146 0 0 −2 18 −17 17 0 0 0 1 −1 0

Nc=1(λ(2),λ(1)),g,β

6. Summary and discussions

In this paper, we have briefly presented framed link invariants inSO(N) Chern–Simonstheory. Then, we studied the expectation value of the observables in topological string theory car-rying SOholonomy. We had derived modified Frobenius equations leading to new polynomialinvariants as a reformulation of framed link invariants inSO(N) Chern–Simons gauge theory.We have proposed new conjectures which are generalisation of the Ooguri–Vafa conjecture andBouchard–Florea–Marino conjecture involving reformulatedSO(N) framed link invariants. Wehave explicitly computed the reformulated polynomial invariants and BPS integer coefficients,corresponding to cross-capc = 1 unoriented topological string amplitudes, for some non-trivialframed knots and framed two-component links verifying the conjecture.

It is still a challenging problem of obtaining cross-capc = 2 unoriented string amplitude onan orientifold of a Calabi–Yau background. This requires deriving the amplitude on a coveringgeometry[17].

Another open question is to studySO(N) Chern–Simons free-energy at largeN for three-manifolds other thanS3. In particular, we have to pose new duality conjectures involvingtopological strings on orientifold background corresponding toSO(N) Chern–Simons theoryon orbifolds ofS3. We hope to study these challenging issues in future.

Acknowledgements

P.B. would like to thank CSIR for the grant. The work of P.R. is supported by Department ofScience and Technology grant under “SERC FAST TRACK Scheme for Young Scientists”.


Appendix A. Knot and link invariants VΛR1 ,ΛR2 ,...,ΛRr[L](q,λ)

In this appendix we present the knot and link invariants for some knots and links with arbitraryframings.

A.1. Unknot with framingp

(A.1)Vλ(1) = (−1)pλp/2[1+ q1/2λ−1/2(−1+ λ)

−1+ q

],

(A.2)V2λ(1) = qpλp(−1+ λ)(−q + q2λ + q1/2λ1/2(−1+ q2))

(−1+ q)2λ(1+ q),

(A.3)Vλ(2) = q−pλp(−1+ λ)(−q2 + qλ + q1/2λ1/2(−1+ q2))

(−1+ q)2λ(1+ q),

(A.4)

V3λ(1) = (−1)pq3pλ3p/2

(−1+ q)3(1+ q)(1+ q + q2)

[qλ−3/2(−1+ q)

(q1/2 + q5/2λ2

− q1/2λ(1+ q2)− λ1/2(−1+ q3)+ λ3/2(−1+ q3))]

(A.5)Vλ(1)+λ(2) = (−1)pλ3p/2(−q + λ)(−1+ qλ)(−q3/2 + q3/2λ + λ1/2(−1+ q3))

(−1+ q)3λ3/2(1+ q + q2),

(A.6)

Vλ(3) = (−1)pq−3pλ3p/2

(−1+ q)3(1+ q)(1+ q + q2)

[qλ−3/2(−q + λ)

(q5/2 + q1/2λ2

− q1/2λ(1+ q2)− λ1/2(−1+ q3)+ λ3/2(−1+ q3))],

(A.7)

V4λ(1) = q6pλ2p

(−1+ q)4λ2(1+ q)2(1+ q2)(1+ q + q2)

[(−1+ λ)(−1+ qλ)

× (−1+ q2λ)(−q2 + q5λ + q3/2λ1/2(−1+ q4))],

(A.8)

V2λ(1)+λ(2) = q2pλ2p

(−1+ q)4(1+ q)2(1+ q2)

[q1/2λ−2(−q1/2 + λ1/2)(q1/2 + λ1/2)

× (q3/2 + λ1/2)(−1+ λ)

(−1+ q2λ)(−1+ q5/2λ1/2)],

(A.9)

V2λ(2) = λ2p

(−1+ q)4λ2(1+ q)2(1+ q + q2)

[q4 + q4λ4 − q3λ3(1+ q)2

− q3λ(1+ q)2 − (−1+ q)q5/2λ1/2(1+ q)2 + (−1+ q)q5/2λ7/2(1+ q)2

+ (−1+ q)q1/2λ3/2(1+ q)2(1+ (−1+ q)q)(

1+ q + q2)− (−1+ q)q1/2λ5/2(1+ q)2(1+ (−1+ q)q

)(1+ q + q2)

− λ2(1+ q + q2)(1+ q(−2+ q

(−1+ (−2+ q)(−1+ q)q(1+ q))))]

,


(A.10)

Vλ(1)+λ(3) = q−2pλ2p

(−1+ q)4(1+ q)2(1+ q2)

[q1/2λ−2(q5/2 + λ1/2)(−1+ λ)

× (−q2 + λ)(−1+ q1/2λ1/2)(1+ q1/2λ1/2)(−1+ q3/2λ1/2)],

(A.11)

Vλ(4) = q−6pλ2p

(−1+ q)4λ2(1+ q)2(1+ q2)(1+ q + q2)

[(−1+ λ)(−q + λ)

× (−q2 + λ)(−q5 + q2λ + q3/2λ1/2(−1+ q4))].

A.2. Trefoil knot with framingp

(A.12)

Vλ(1) = (−1)p

(−1+ q)2(1+ q)

[q−2λ(p+1)/2(q3/2 − q11/2 + q6λ1/2 − q3/2λ

+ qλ1/2 + q3λ3/2 + q5λ5/2 − q5/2λ − q6λ3/2 + q9/2λ + q11/2λ

− q3λ1/2 − q4λ1/2 − qλ3/2 + q4λ3/2 + q5/2λ2 − q9/2λ2 + q2λ5/2

− q3λ5/2 − q4λ5/2)],

(A.13)

V2λ(1) = 1

(−1+ q)2(1+ q)

[qp−3λp+1(q2 + q5 + q6 + q8 + (−1+ q)2q9λ5

× (1+ q) − q19/2λ9/2(−1+ q2)+ q10λ4(1+ q − q3)− q2λ(1+ q)

× (1+ q2)(1+ q3 + q4)+ (−1+ q)q3/2λ3/2(1+ q)

(1+ (−1+ q)q

)× (

1+ q + q2)(1+ q + q2 + q3 + q4)− (−1+ q)q5/2λ5/2(1+ q)

× (1+ q2)(1+ q + q2 + q3 + q4 + q5 + q6)+ q5λ3(1+ q)

× (−1− q2 − q4 − q6 + q7)− q3/2λ1/2(−1− q3 + q7 + q8)+ q9/2λ7/2(−1− q3 + q7 + q8)− q3λ2(1+ (−1+ q)q

)× (−1+ q

(−2+ q(1+ q2)(−3− 3q + q3))))],

(A.14)

Vλ(2) = 1

(−1+ q)2(1+ q)

[q−(21/2+p)λp+1((−1+ q)2q9/2λ5(1+ q)

− q5λ9/2(−1+ q2)+ q7/2λ4(−1+ q2 + q3)− q15/2λ(1+ q)

× (1+ q2)(1+ q + q4)+ (−1+ q)q5λ3/2(1+ q)

(1+ (−1+ q)q

)× (

1+ q + q2)(1+ q + q2 + q3 + q4)+ q17/2(1+ q2 + q3 + q6)− (−1+ q)q4λ5/2(1+ q)

(1+ q2)(1+ q + q2 + q3 + q4 + q5 + q6)

− q7/2λ3(1+ q)(−1+ q + q3 + q5 + q7)− q7λ1/2(−1− q + q5 + q8)

+ q4λ7/2(−1− q + q5 + q8)+ q9/2λ2(−1+ q

(1+ q

(1+ q

(1+ q + q2)(1+ q + q2 + q4)))))].


A.3. Hopf link with framingp1 on first strand andp2 on the second

(A.15)

Vλ(1),λ(1) = 1

(−1+ q)2

[(−1)p1+p2q−1/2λ(p1+p2−2)/2(q1/2 + λ1/2)

× (−1+ q1/2λ1/2)(−1+ q − q2 + (−1+ q)q1/2λ1/2

+ λ(1+ (−1+ q)q

))],

(A.16)

V2λ(1),λ(1) = 1

(−1+ q)3(1+ q)

[(−1)p2q−1+p1λ(2p1+p2−3)/2(−1+ λ1/2)

× (1+ λ1/2)(q1/2 + λ1/2)(−1+ q1/2λ1/2)(−1+ q3/2λ1/2)

× (1− q + q3 + q1/2λ1/2(1+ (−1+ q)q2))],

(A.17)

Vλ(2),λ(1) = 1

(−1+ q)3(1+ q)

[(−1)p2q−(2+p1)λ(2p1+p2−3)/2

× (−(q7/2(1+ (−1+ q)q2))+ q3/2λ3(1− q + q3)+ q5/2λ

(1+ q + q4)+ (−1+ q)q2λ1/2(1+ q + q4)

− q3/2λ2(1+ q3 + q4)+ (−1+ q)qλ5/2(1+ q3 + q4)− qλ3/2(−1+ q6))],

(A.18)

V3λ(1),λ(1) = 1

(−1+ q)4(1+ q)(1+ q + q2)

[(−1)3p1+p2q−3+3p1λ(3p1+p2−4)/2

× (−1+ λ)(q7λ3(1+ (−1+ q)q3)− q2(1− q + q4)

− q5λ2(1+ q + q5)+ q3λ(1+ q4 + q5)

+ (−1+ q)q9/2λ5/2(1+ q + q2 + q6)+ q3/2λ1/2(−1+ q − q4 + q7)− q5/2λ3/2(−1+ q8))],

(A.19)

Vλ(1)+λ(2),λ(1) = 1

(−1+ q)4(1+ q)2(1+ q + q2)

[(−1)3p1+p2q−2λ(3p1+p2−4)/2

× (q3 + q4 + q8 + q9 − q2λ(1+ q)2(1+ q2)(1+ q4)

− q2λ3(1+ q)2(1+ q2)(1+ q4)+ (−1+ q)q3/2λ7/2(1+ q)2(1+ q2)(1+ q4)+ q2λ2(1+ q)2(1+ q + q2)(1+ q4)+ (−1+ q)q1/2λ3/2(1+ q)2(1+ q2)(1+ q + q2)(1+ q4)− (−1+ q)q1/2λ5/2(1+ q)2(1+ q2)(1+ q + q2)(1+ q4)+ q3λ4(1+ q + q5 + q6)− q3/2λ1/2(−1− q + q8 + q9))],


(A.20)

Vλ(3),λ(1) = 1

(−1+ q)4(1+ q)(1+ q + q2)

[(−1)3p1+p2q−2−3p1λ(−4+3p1+p2)/2

× (−1+ λ)(−(q6(1+ (−1+ q)q3))+ qλ3(1− q + q4)

+ q4λ(1+ q + q5)− q2λ2(1+ q4 + q5)

+ (−1+ q)q7/2λ1/2(1+ q + q2 + q6)+ q1/2λ5/2(−1+ q − q4 + q7)− q3/2λ3/2(−1+ q8))],

(A.21)

V2λ(1),2λ(1) = 1

(−1+ q)4(1+ q)2

[q−4+p1+p2λ−2+p1+p2

× (−(q2λ(1+ q)(1− q + q3 + q6))

− (−1+ q)q3/2λ1/2(1+ q)(1− q + q3 + q6)

− q5λ3(1+ q)(1+ q3 − q5 + q6)

+ (−1+ q)q9/2λ7/2(1+ q)(1+ q3 − q5 + q6)

− (−1+ q)q5/2λ5/2(1+ q)(1+ q2 + q3 + q4 + q5 + q8)

+ q3λ2(1+ q3 + 2q4 + q5 + q8)+ (−1+ q)q3/2λ3/2(1+ q)

(1+ q3 + q4 + q5 + q6 + q8)

+ q6λ4(1+ (−1+ q)q(1+ q)(1+ (−1+ q)q2))

+ q2(1+ (−1+ q)q(1+ q)(1+ q − q2 + q3)))],

(A.22)

V2λ(1),λ(2) = 1

(−1+ q)4(1+ q)2

[q−2+p1−p2λ−2+p1+p2(−1+ λ)

(−q3 + q5 − q7

− (−1+ q)q1/2λ3/2(1+ q)2(1+ (−1+ q)q)(

1+ q4)+ q3λ3(1− q2 + q4)− q2λ2(1+ q3 + q6)+ q2λ

(1+ q3 + q6)

+ q3/2λ1/2(−1+ q2 − q5 + q7)+ (−1+ q)q3/2λ5/2(1+ q)2(1+ (−1+ q)q

(1+ q2)))],

(A.23)

Vλ(2),λ(2) = 1

(−1+ q)4(1+ q)2

[q−9/2−p1−p2λ−2+p1+p2

× (−(q5/2λ3(1+ q)(1− q + q3 + q6))

+ (−1+ q)q2λ7/2(1+ q)(1− q + q3 + q6)

− (−1+ q)q5λ1/2(1+ q)(1+ q3 − q5 + q6)

− q11/2λ(1+ q)(1+ q3 − q5 + q6)

+ (−1+ q2)q3λ3/2(1+ q2 + q3 + q4 + q5 + q8)+ q7/2λ2(1+ q3 + 2q4 + q5 + q8)− (−1+ q)q2λ5/2(1+ q)

(1+ q3 + q4 + q5 + q6 + q8)

+ q13/2(1+ (−1+ q)q(1+ q)(1+ (−1+ q)q2))

+ q5/2λ4(1+ (−1+ q)q(1+ q)(1+ q − q2 + q3)))].


Appendix B. SO(N) reformulated invariants gR1,R2,...,Rr (q,λ)

B.1. Trefoil with framingp

(B.1)

gλ(1) (q, λ) = 1

(−1+ q)q

[(−1)pλ(1+p)/2(−(q1/2(1+ q2))+ (−1+ q3)λ1/2

+ q1/2(1+ q + q2)λ − (−1+ q3)λ3/2 − q3/2λ2

+ (−1+ q)qλ5/2)],

(B.2)

g2λ(1) (q, λ) = 1

2(−1+ q)2q2(1+ q)

[2(−1+ q)2q2(1+ q)

+ λ1+p(−2qp

(−(q + q4 + q5 + q7)+ q1/2(−1− q3 + q7 + q8)λ1/2

+ q(1+ q)(1+ q2)(1+ q3 + q4)λ

− (−1+ q)q1/2(1+ q)(1+ (−1+ q)q

)(1+ q + q2)

× (1+ q + q2 + q3 + q4)λ3/2

+ q2(1+ (−1+ q)q)(−1+ q

(−2+ q(1+ q2)(−3− 3q + q3)))λ2

+ (−1+ q)q3/2(1+ q)(1+ q2)(1+ q + q2 + q3 + q4 + q5 + q6)λ5/2

− q4(1+ q)(−1− q2 − q4 − q6 + q7)λ3

− q7/2(−1− q3 + q7 + q8)λ7/2 + q9(−1− q + q3)λ4

+ q17/2(−1+ q2)λ9/2 − (−1+ q)2q8(1+ q)λ5)+ (−1)p

(−((−1)p(1+ q)(q1/2(1+ q2)− (−1+ q3)λ1/2

− q1/2(1+ q + q2)λ + (−1+ q3)λ3/2 + q3/2λ2 − (−1+ q)qλ5/2)2)+ (−1+ q)(q + λ)(−1+ qλ)

× (−1+ (q − λ)(−λ − qλ2 + q3(−1+ λ2)))))],

gλ(2) (q, λ) = 1

2(−1+ q)2(1+ q)

[q−7−pλ1+p

(2(q5 + q7 + q8 + q11

− q7/2(−1− q + q5 + q8)λ1/2 − q4(1+ q)(1+ q2)(1+ q + q4)λ

+ (−1+ q)q3/2(1+ q)(1+ (−1+ q)q

)(1+ q + q2)

× (1+ q + q2 + q3 + q4)λ3/2

+ q(−1+ q

(1+ q

(1+ q

(1+ q + q2)(1+ q + q2 + q4))))λ2

− (−1+ q)q1/2(1+ q)(1+ q2)(1+ q + q2 + q3 + q4 + q5 + q6)λ5/2

− (−1+ q + q3 + q5 + q7)λ3(1+ q) + q1/2(−1− q + q5 + q8)λ7/2

+ (−1+ q2 + q3)λ4 − q3/2(−1+ q2)λ9/2 + (−1+ q)2

q(1+ q)λ5)− (−1)pq5+p

((−1)p(1+ q)

(q1/2(1+ q2)− (−1+ q3)λ1/2

− q1/2(1+ q + q2)λ + (−1+ q3)λ3/2 + q3/2λ2 − (−1+ q)qλ5/2)2


+ (−1+ q)(q + λ)(−1+ qλ)

(B.3)× (−1+ (q − λ)(−λ − qλ2 + q3(−1+ λ2)))))].

B.2. Connected sum of trefoil and trefoil with framingp

gλ(1) (q, λ) = 1

(−1+ q)q2

[(−1)pλ(3+p)/2(q1/2 + λ1/2)(−1+ q1/2λ1/2)

(B.4)× (−1+ (q1/2 − λ1/2)(q1/2(q(−1+ λ) − λ

)− λ1/2))2],g2λ(1) (q, λ) = 1

2(−1+ q)2q4(1+ q)

[2(−1+ q)2q4(1+ q)

+ λ3+p(−((−1+ (

q1/2 − λ1/2)(q1/2(q(−1+ λ) − λ)− λ1/2))4

× (1+ q)(q1/2 + λ1/2)2(−1+ q1/2λ1/2)2)

+ 2q1/2+p(q1/2 + λ1/2)(−1+ q3/2λ1/2)(−1+ λ)

[(1+ q3(1+ q + q3)

− (−1+ q)q3/2(1+ q)2(1+ (−1+ q)q)λ1/2

− q(1+ q)2(1− q + 2q3 − 2q4 + q5)λ+ (−1+ q)q5/2(1+ q)

(1+ q − q2 + q3 + q4)λ3/2

+ q3(1+ (−1+ q)q(1+ q)(1+ (−1+ q)2q

))λ2

− q9/2(−1+ q + q2 − 2q3 + q5)λ5/2 + (−1+ q)2q6(1+ q)λ3)]2− (−1)p(−1+ q)(q + λ)(−1+ qλ)

(B.5)× (−1+ (q − λ)(−λ − qλ2 + q3(−1+ λ2)))2)],

gλ(2) (q, λ) = 1

2(−1+ q)2(1+ q)

[q−57/2−pλ3+p

(2q14(−1+ λ1/2)(1+ λ1/2)

× (q3/2 + λ1/2)(−1+ q1/2λ1/2)[(q7/2(1+ q2 + q3 + q6)

− (−1+ q)q3(1+ q)2(1+ (−1+ q)q)λ1/2

− q3/2(1+ q)2(1+ (−1+ q)q)(

1− q + q3)λ+ (−1+ q)q(1+ q)

(1+ q − q2 + q3 + q4)λ3/2

+ q1/2(1+ (−1+ q)q(1+ q)2(2+ (−2+ q)q))

λ2

+ (1+ q2(−2+ q + q2 − q3))λ5/2 + (−1+ q)2q1/2(1+ q)λ3)]2

+ q49/2+p(−((−1+ (

q1/2 − λ1/2)(q1/2(q(−1+ λ) − λ)− λ1/2))4

× (1+ q)(q1/2 + λ1/2)2(−1+ q1/2λ1/2)2)

+ (−1)p(−1+ q)(q + λ)(−1+ qλ)

(B.6)× (−1+ (q − λ)(−λ − qλ2 + q3(−1+ λ2)))2))].


B.3. Hopf link with framingp1 on first strand andp2 on the second

gλ(1),λ(1) (q, λ) = (−1)p1+p2q−1/2(q1/2 + λ1/2)(−1+ q1/2λ1/2)(B.7)× (−1+ λ)λ

(−2+p1+p2)/2,

g2λ(1),λ(1) (q, λ) = 1

(−1+ q)3q(1+ q)

[(−1)p2

(−((−1+ q)2q1/2(1+ q)

× (q1/2 + λ1/2)(−1+ q1/2λ1/2))+ qp1

(−(q2 + q3/2λ1/2)+ (

1− q + q3 + q1/2(1+ (−1+ q)q2)λ1/2))(−1+ q3/2λ1/2))(B.8)× (

q1/2 + λ1/2)(−1+ q1/2λ1/2)(−1+ λ)λ(−3+2p1+p2)/2],gλ(2),λ(1) (q, λ) = −1

(−1+ q)3(1+ q)

[(−1)p2q−1−p1

((−1+ q)2q1/2+p1(1+ q)

× (q1/2 + λ1/2)(−1+ q1/2λ1/2)

+ (q3/2 + λ1/2)(−q3/2 + (

q1/2(1+ (−1+ q)q2)+ (−1+ q − q3)λ1/2)+ q2λ1/2))

(B.9)× (q1/2 + λ1/2)(−1+ q1/2λ1/2)(−1+ λ)λ(−3+2p1+p2)/2],

g3λ(1),λ(1) (q, λ) = 1

(−1+ q)2q5/2(1+ q)

[(−1)p1+p2

(q1/2 + λ1/2)(−1+ q1/2λ1/2)

× (−1+ λ)λ(−4+3p1+p2)/2(q2(1+ q)(q1/2 + λ1/2)2(−1+ q1/2λ1/2)2

+ q1+3p1(1+ q1/2λ1/2)(−1+ qλ1/2)(1+ qλ1/2)(−1+ q5/2λ1/2)

(B.10)

− q3/2+p1(q1/2 + λ1/2)(−1+ q3/2λ1/2)(−1− 2q + q(2+ q)λ

))],

gλ(1)+λ(2),λ(1) (q, λ) = −1

(−1+ q)2(1+ q)

[(−1)p1+p2q−3/2−p1

(q1/2 + λ1/2)

× (−1+ q1/2λ1/2)(−1+ λ)λ(−4+3p1+p2)/2

× (−(q1/2(q3/2 + λ1/2)(−1+ q1/2λ1/2)(q(2+ q − 2λ) − λ))

+ q1/2+2p1(q1/2 + λ1/2)(−1+ q3/2λ1/2)(−1− 2q + q(2+ q)λ

)+ qp1(1+ q)

(−3q2 + (−1+ q)q1/2(1+ q(4+ q))λ1/2

+ (1+ q

(−3+ q(10+ (−3+ q)q

)))λ

(B.11)− (−1+ q)q1/2(1+ q(4+ q))λ3/2 − 3q2λ2))],

gλ(3),λ(1) (q, λ) = 1

(−1+ q)2(1+ q)

[(−1)p1+p2q−3/2−3p1(−1+ q1/2λ1/2)(−1+ λ)

× λ(−4+3p1+p2)/2(q1+3p1(1+ q)(q1/2 + λ1/2)3(−1+ q1/2λ1/2)2


+ q1/2+2p1(q1/2 + λ1/2)(q3/2 + λ1/2)(−1+ q1/2λ1/2)(B.12)

× (q(2+ q − 2λ) − λ

)+ (q2 − λ

)(q7/2 + qλ1/2 − q5/2λ − λ3/2))],

g2λ(1),2λ(1) (q, λ) = 1

2(−1+ q)4q(1+ q)

[λ−2+p1+p2((q1/2 + λ1/2)2(−1+ q1/2λ1/2)2

× (2q3/2+p2(q1/2 + λ1/2)(−1+ q3/2λ1/2)(−1+ λ)

+ 2qp1(−1+ λ)(−q2 + q3/2(−1+ q2)λ1/2 + q3λ

)− (1+ q)

(−1− (−5+ q)q − 3(−1+ q)q1/2λ1/2

+ (1+ (−5+ q)q

)λ)(−1+ q − q2

+ (−1+ q)q1/2λ1/2 + (1+ (−1+ q)q

)λ))

+ (−1)p1+p2(q + 1)(−3(−1)p1+p2q

(q1/2 + λ1/2)4(−1+ q1/2λ1/2)4

− (−1+ q)4(q + λ)(−1+ qλ)(−1+ λ2))

+ 2(−1+ λ1/2)(1+ λ1/2)(q1/2 + λ1/2)

× (−1+ q1/2λ1/2)(−1+ q3/2λ1/2)(−(qp1 + qp2)(q1/2 + λ1/2)× (−1+ q1/2λ1/2)(1− q + q3 + q1/2(1+ (−1+ q)q2)λ1/2)+ q(q + 1)(−1+ q)2q−(5/2)+p1+p2(1+ q1/2λ1/2)

(B.13)

× (−1+ q(1− q + (−1+ q)q1/2λ1/2 + q

(1+ (−1+ q)q

)λ))))]

,

g2λ(1),λ(2) (q, λ) = 1

2(−1+ q)4q(q + 1)

[λ−2+p1+p2((−1)p1+p2(q + 1)

× (−3(−1)p1+p2q(q1/2 + λ1/2)4(−1+ q1/2λ1/2)4

+ (−1+ q)4(q + λ)(−1+ qλ)(−1+ λ2))

+ (q1/2 + λ1/2)2(−1+ q1/2λ1/2)2(−(q + 1)

× (−1− (−5+ q)q − 3(−1+ q)q1/2λ1/2 + (1+ (−5+ q)q

)λ)

× (−1+ q − q2 + (−1+ q)q1/2λ1/2 + (1+ (−1+ q)q

)λ)

+ 2q(−1+ λ)(q1/2−p2(q3/2 + λ1/2)(−1+ q1/2λ1/2)

+ qp1(−q + q1/2(−1+ q2)λ1/2 + q2λ)))

+ 2q−(3/2)−p2(q1/2 + λ1/2)(−1+ q1/2λ1/2)× (−q(q + 1)(−1+ q)2qp1(q1/2 − λ1/2)(−1+ λ1/2)(1+ λ1/2)× (

q3/2 + λ1/2)(1+ q1/2λ1/2)(−1+ q3/2λ1/2)− q3/2(q1/2 + λ1/2)(−1+ q1/2λ1/2)(−1+ λ)

(−q2 + q4 − q5

+ λ1/2(−q1/2 + q3/2 − q7/2 + q9/2)+ λ(1− q + q3)


+ qp1+p2(−1+ q − q3 + q1/2(−1+ q − q3 + q4)λ1/2

(B.14)+ q2(1+ (−1+ q)q2)λ))))],gλ(2),λ(2) (q, λ) = 1

2(−1+ q)4(1+ q)

[q−(9/2)−p1−p2λ−2+p1+p2(2(−1+ q)2q2(1+ q)

× (q1/2 − λ1/2)(−1+ λ1/2)(1+ λ1/2)(q1/2 + λ1/2)(q3/2 + λ1/2)

× (−1+ q1/2λ1/2)(q2 − q3 + q4 +q3/2+p1(q1/2 + λ1/2)(−1+ q1/2λ1/2)+ q3/2λ1/2 − q5/2λ1/2 − λ + qλ − q2λ

)+ qp2(1+ q)

(2(−1+ q)2q7/2(q1/2 + λ1/2)2(−1+ q1/2λ1/2)2

× (q2 − (−1+ q)q1/2λ1/2 − λ

)(−1+ λ)

− q7/2+p1(−1+ q1/2λ1/2)(3(−1)2(p1+p2)q(q1/2 + λ1/2)4

× (−1+ q1/2λ1/2)3+ (

q1/2 + λ1/2)2(−1+ q1/2λ1/2)(−1− (−5+ q)q

− 3(−1+ q)q1/2λ1/2 + (1+ (−5+ q)q

)λ)(−1+ q − q2 + (−1+ q)

× q1/2λ1/2 + (1+ (−1+ q)q

)λ)

(B.15)+ (−1)p1+p2(−1+ q)4(1+ q1/2λ1/2)(q + λ)(−1+ λ2))))].

B.4. Connected sum of trefoil and Hopf link with framingp1 on trefoil and framingsp1 andp2

on the Hopf link

(B.16)

gλ(1),λ(1) (q, λ) = q−3/2[(−1)p1+p2(−1+ λ)λ(p1+p2)/2(−(q1/2(1+ q2))+ (−1+ q3)λ1/2 + q1/2(1+ q + q2)λ − (−1+ q3)λ3/2

− q3/2λ2 + (−1+ q)qλ5/2)],

(B.17)

g2λ(1),λ(1) (q, λ) = −1

(−1+ q)q3

[(−1)p2

(−1+ λ1/2)(1+ λ1/2)(q1/2 + λ1/2)× (−1+ q1/2λ1/2)λ(1+2p1+p2)/2(q1/2[(−1+ (

q1/2 − λ1/2)× (

q1/2(q(−1+ λ) − λ)− λ1/2))]2(q1/2 + λ1/2)(−1+ q1/2λ1/2)

+ qp1(1+ q3 + q4 + q6 + q1/2(1− q4(−1+ q + q3))λ1/2

− q(1+ q + 2q3 + q4 + q5 + q6 + q7)λ

+ q3/2(−1− q + q7 + q8)λ3/2 + q3(1+ q2 + 2q3 + q6)λ2

+ q7/2(1+ q3 − q4 + q6 − 2q7)λ5/2 + q8(−1+ (−1+ q)q)λ3

+ q15/2(−1+ q − q3 + q4)λ7/2 − (−1+ q)2q8(1+ q)λ4))],


(B.18)

gλ(2),λ(1) (q, λ) = −1

−1+ q

[(−1)p2q−17/2−p1

(q1/2 + λ1/2)(−1+ q1/2λ1/2)(−1+ λ)

× λ(1+2p1+p2)/2(q11/2 + q15/2 + q17/2 + q23/2 + q6+p1

× (−1+ (q1/2 − λ1/2)(q1/2(q(−1+ λ) − λ

)− λ1/2))2× (

q1/2 + λ1/2)(−1+ q1/2λ1/2)+ λ1/2(q4 + q6 − q7 − q11)+ λ

(−q7/2 − q9/2 − q11/2 − q13/2 − 2q15/2 − q19/2 − q21/2)+ λ3/2(−q2 − q3 + q9 + q10)+ λ2(q5/2 + 2q11/2 + q13/2 + q17/2)+ λ5/2(2q − q2 + q4 − q5 − q8)+ λ3(q3/2 − q5/2 − q7/2)+ λ7/2(−1+ q − q3 + q4)+ λ4(−q1/2 + q3/2 + q5/2 − q7/2))].

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SU(N) quantum Racah coefficients and non-torus links

Zodinmawia, P. Ramadevi ∗

Department of Physics, Indian Institute of Technology Bombay, Mumbai, 400076, India

Received 3 October 2012; accepted 31 December 2012

Available online 11 January 2013

Abstract

It is well known that the SU(2) quantum Racah coefficients or the Wigner 6j symbols have a closed formexpression which enables the evaluation of any knot or link polynomials in SU(2) Chern–Simons fieldtheory. Using isotopy equivalence of SU(N) Chern–Simons functional integrals over three-balls with oneor more S2 boundaries with punctures, we obtain identities to be satisfied by the SU(N) quantum Racahcoefficients. This enables evaluation of the coefficients for a class of SU(N) representations. Using thesecoefficients, we can compute the polynomials for some non-torus knots and two-component links. Theseresults are useful for verifying conjectures in topological string theory.© 2013 Elsevier B.V. All rights reserved.

Keywords: Chern–Simons field theory; Knot polynomials; Ooguri–Vafa conjecture

1. Introduction

Following the seminal work of Witten [1] on Chern–Simons theory as a theory of knots andlinks, generalised invariants [2,3] for any knot or link can be directly obtained without goingthrough the recursive procedure. For torus links, which can be wrapped on a two-torus T 2, usingthe torus link operators [4], explicit polynomial form of these invariants could be obtained. How-ever for non-torus links, the generalised invariants [2] in SU(N) Chern–Simons theory involvesSU(N) quantum Racah coefficients which are not known in closed form as known for SU(2)

[5,6]. This prevents in writing the polynomial form for the non-torus links.

For simple SU(N) representations placed on knots, whose Young Tableau are , and ,we had obtained some Racah coefficients from isotopy equivalence of knots and links which

* Corresponding author.E-mail addresses: [email protected] (Zodinmawia), [email protected] (P. Ramadevi).

0550-3213/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nuclphysb.2012.12.020

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206 Zodinmawia, P. Ramadevi / Nuclear Physics B 870 [PM] (2013) 205–242

Fig. 1. Plat representation for some non-torus knots.

was useful to obtain polynomial for few non-torus knots [2,3]. Going beyond these simple rep-resentations and finding the quantum Racah coefficients appeared to be a formidable task. Infact, determining these coefficients would help in verifying the topological string conjectures fora general non-torus knot or link proposed by Ooguri–Vafa [7] and Labastida–Marino–Vafa [8].Using the few Racah coefficients data [2], Ooguri–Vafa conjecture for 41,61 non-torus knots asindicated in Fig. 1 were verified [9]. For verifying Labastida–Marinoi–Vafa conjecture for thenon-torus two-component links [8], we need to evaluate the non-torus link whose componentknots carry different representations.

These non-torus links invariants will also be useful to generalise some of the results of recentpapers [10–13] where torus knots and links are studied. So, it is very important to determine theSU(N) quantum Racah coefficients.

Using the correspondence between Chern–Simons functional integral and the correlator con-formal blocks states in the corresponding Wess–Zumino conformal field theory [1], we can deriveidentities to be obeyed by the SU(N) quantum Racah coefficients. Using the identities and the

Zodinmawia, P. Ramadevi / Nuclear Physics B 870 [PM] (2013) 205–242 207

Fig. 2. Plat representation for some non-torus links.

properties of quantum dimensions for any N , we could determine the form of these coefficientsfor some class of SU(N) representations. These coefficients are needed to obtain polynomialinvariants of some non-torus knots and non-torus two-component links.

The plan of the paper is as follows: In Section 2, we briefly review Chern–Simons functionalintegrals and properties of the Racah coefficients. In Section 3, we systematically study the equiv-alence of states and obtain identities to be obeyed by the SU(N) quantum Racah coefficients. InSection 4, we tabulate the SU(N) quantum Racah coefficients. In Appendix A we give the gen-eralised Chern–Simons invariant for the non-torus knots and non-torus links in Figs. 1 and 2.Then we present the polynomial form of these invariants for few representations in Appendix Band Appendix C. We also verify Ooguri–Vafa conjecture for knots and Labastida–Marino–Vafaconjecture for links in Appendix D. In the concluding section, we summarise and discuss someof the open problems.

2. Chern–Simons field theory

Chern–Simons fields on S3 with U(1) × SU(N) gauge group with levels k1, k respectively isgiven by the following action:

S = k1

4π

∫S3

B ∧ dB + k

4π

∫S3

Tr

(A ∧ dA + 2

3A ∧ A ∧ A

), (2.1)

where B is the U(1) gauge connection and A is the SU(N) matrix valued gauge connection. Theobservables in this theory are Wilson loop operators:

W(R1,n1),(R2,n2),...,(Rr ,nr )[L] =r∏

β=1

TrRβ UA[Kβ ]Trnβ UB [Kβ ], (2.2)

where the holonomy of the gauge field A around a component knot Kβ , carrying a representationRβ , of a r-component link is denoted by UA[Kβ ] = P [exp

∮Kβ

A] and nβ is the U(1) charge


carried by the component knot Kβ . The expectation value of these Wilson loop operators are thelink invariants:

V{U(N)}(R1,n1),...

[L](q,λ) = ⟨W(R1,n1),...[L]⟩= ∫ [DB][DA]eiSW(R1,n1),...,(Rr ,nr )[L]∫ [DB][DA]eiS

= V{SU(N)}R1,R2,...,Rr

[L]V {U(1}n1,n2,...,nr

[L]. (2.3)

We make a specific choice of the U(1) charges and the coupling k1 so that the above invariant isa polynomial in two variables q = exp( 2πi

k+N), λ = qN [14,15]

nβ = lβ√N

, k1 = k + N, (2.4)

where lβ is the total number of boxes in the Young Tableau representation Rβ . The U(1) linkinvariant for the link with this substitution gives

V{U(1}l1√N

,..., lr√N

[L] = (−1)∑

β lβpβ exp

(iπ

k + N

r∑β=1

l2βpβ

N

)exp

(iπ

k + N

∑α �=β

lαlβ lkαβ

N

), (2.5)

where pβ is the framing number of the component knot Kβ and lkαβ is the linking numberbetween the component knots Kα and Kβ . In order to directly evaluate SU(N) link invariants,we need to use the following two ingredients:

1. The relation between SU(N) Chern–Simons functional integral on the three-dimensionalball to the two-dimensional SU(N)k Wess–Zumino conformal field theory on the boundaryof the three-ball [1].

2. Any knot or link can be drawn as a platting of braids [16].

In Fig. 1 and Fig. 2 we have drawn some non-torus knots and non-torus links as a plat representa-tion of braids. We have labelled them in the Thistlethwaite notation and written their braid words.We have indicated the orientation and labelled the representation Ri on the component knots inthe link. Note that b

(−)i ({b(−)

i }−1) in the braid word denotes right-handed crossing (left-handed

crossing) between i-strand and (i + 1)-th strand which are antiparallelly oriented. Similarly b(+)j

({b(+)j }−1) denotes right-handed crossing (left-handed crossing) between j and (j + 1)-th strand

which are parallelly oriented. The plat representation of these non-torus knots and non-toruslinks involves braids with four-strands. Hence we can view these knots and links in S3 as gluingof three-balls with 4-punctured boundary as shown in Fig. 3. There are two three-balls B1 andB3 with one S2 boundary. A three-ball denoted as B2 in Fig. 3 with two S2 boundaries with abraid B which can represent any of the braid words corresponding to the non-torus knots andlinks in Figs. 1 and 2. The gluing of the three-balls are along the oppositely oriented S2 bound-aries. The functional integral over the ball B3 is given by a state |χ(2)

0 〉 where the superscriptdenotes the label of the S2 boundary. The representation Ri indicate that the lines are going intothe S2 boundary of the three-ball and the conjugate representation denotes the lines going out ofthe S2 boundary. The state corresponding to a functional integral on a three-ball with an oppo-sitely oriented boundary is written in a dual space along with conjugating all the representationsas illustrated for the ball B1. The expectation value of the Wilson loop operator gives the linkinvariant for a non-torus link L


Fig. 3. Link in S3 viewed as gluing of three-balls with four-punctured S2 boundaries.

Fig. 4. Four-point conformal block bases states.

VSU(N)R1,R2

[L] = ⟨Ψ (1)0

∣∣Bν(1),(2)∣∣χ(2)

0

⟩. (2.6)

These invariants multiplied with the U(1) invariant (2.5) are polynomials in two variables q =exp(2πi/(k + N)) and λ = qN . In order to see the polynomial form, we write these states on afour-punctured boundary in a suitable basis of four-point conformal block of the SU(N)k Wess–Zumino conformal field theory. There are two different four-point conformal block bases asshown in Fig. 4 where t ∈ (R1 ⊗ R2) ∩ (R3 × R4) and s ∈ (R2 ⊗ R3) ∩ (R1 × R4). Using thesebases, the states corresponding to three-balls B1, B2 and B3 in Fig. 3, can be expanded as [2,3]⟨

Ψ(1)0

∣∣=√dimq R1 dimq R2⟨φ0(R1,R1,R2, R2)

(1)∣∣

=∑

s∈R1⊗R2

ε(R1,R2)s

√dimq s

⟨φs(R1,R1,R2, R2)

(1)∣∣, (2.7)

Bν(1),(2) = B∣∣φt (R2, R2, R1,R1)

(1)⟩⟨φt (R2,R2,R1, R1)

(2)∣∣

= B∣∣φs(R2, R2, R1,R1)

(1)⟩⟨φs(R2,R2,R1, R1)

(2)∣∣, (2.8)∣∣χ(2)

0

⟩=√dimq R1 dimq R2∣∣φ0(R2, R2, R1,R1)

(2)⟩

=∑

s∈R1⊗R2

ε(R1,R2)s

√dimq s

∣∣φs(R2, R2, R1,R1)(2)⟩, (2.9)


where the subscript ‘0’ in the basis state in Eqs. (2.7), (2.9) denotes the singlet state. The statesare written such that the invariant of a simple circle (also called unknot) carrying representationR is normalised as dimq R, which is the quantum dimension of the representation R, defined as

dimq R =∏α>0

[α · (ρ + ΛR)][α · ρ] , (2.10)

where ΛR denotes the highest weight of the representation R, α’s are the positive roots and ρ isequal to the sum of the fundamental weights of the Lie group SU(N). The square bracket refersto quantum number defined as

[n] = qn2 − q− n

2

q12 − q− 1

2

. (2.11)

The symbol ε(R1,R2)s = ±1 which we will fix from equivalence of states in the next section. To

operate the braid word B in Eq. (2.8), we need to find the eigenbasis of the braiding generatorsb

(±)i ’s.

The conformal block |φt (R1,R2, R3, R4)〉 is suitable for the braiding operator b(±)1 and b

(±)3 .

Similarly braiding in the middle two strands involving the operator b(±)2 requires the conformal

block |φs(R1,R2, R3, R4)〉. That is,

b(±)2

∣∣φs(R1,R2, R3, R4)⟩= λ(±)

s (R2, R3)∣∣φs(R1, R3,R2, R4)

⟩,

b(±)1

∣∣φt (R1,R2, R3, R4)⟩= λ

(±)t (R1,R2)

∣∣φs(R2,R1, R3, R4)⟩,

b(±)3

∣∣φs(R1,R2, R3, R4)⟩= λ

(±)t (R3, R4)

∣∣φs(R1,R2, R4, R3)⟩, (2.12)

where braiding eigenvalues λ(±)t (R1,R2) in vertical framing are

λ(±)t (R1,R2) = ε

(±)t;R1,R2

(q

CR1+CR2

−CRt2

)±1. (2.13)

In this framing, framing number pβ for the component knot is equal to writhe w of that compo-nent knot which is the difference between total number of left-handed crossings and total numberof right-handed crossing. For example, torus knots 41, 52 in Fig. 1 have writhe w equal to 0 and5 respectively. The symbol ε

(±)t;R1,R2

is a sign which can be fixed by studying equivalence of statesor equivalence of links which we shall elaborate for a class of representations in the next sectionand CR denotes the quadratic Casimir for the representation R given by

CR = κR − l2

2N, κR = 1

2

[Nl + l +

∑i

(l2i − 2ili

)], (2.14)

where li is the number of boxes in the i-th row of the Young Tableau representation R and l isthe total number of boxes. The two bases in Fig. 4 are related by a duality matrix a as follows:∣∣φt (R1,R2, R3, R4)

⟩= ats

[R1 R2R3 R4

] ∣∣φs(R1,R2, R3, R4)⟩. (2.15)

From the definition of t , s, we can see that the duality matrix a obeys the following properties:

ats

[R1 R2R3 R4

]= ast

[R3 R2R1 R4

]= ast

[R1 R4R3 R2

]= at s

[R3 R4R1 R2

]= ats

[R3 R4R1 R2

]. (2.16)


If one of the representation is singlet (denoted by 0), we see that the matrix elements are

ats

[R1 0R3 R4

]= δtR1δsR3

, ats

[0 R2R3 R4

]= δtR2δsR4,

ats

[R1 R20 R4

]= δtR4δsR2, ats

[R1 R2R3 0

]= δtR3δsR1

. (2.17)

From the procedure presented in this section, we can write the U(N) invariants of non-torusknots and links as a product of U(1) invariant times SU(N) invariant (2.6). For example, non-torus knot 52 with framing p = 5, the invariant will be

V{U(N)}R [52;p] = (−1)5lq

5l22N

∑s,t,s′

εR,Rs

√dimq sε

R,Rs′√

dimq s′(λ(+)s (R,R)

)−1

× ats

[R R

R R

](λ

(−)t (R,R)

)−2ats′[

R R

R R

](λ

(+)

s′ (R,R))−2

, (2.18)

where l is the total number of boxes in the Young Tableau representing R and we have indi-cated the framing number of the knot. We could add additional framing p1 by multiplying theseinvariants by a framing factor as follows:

V{U(N)}R

[K; (p + p1)

]= (−1)lp1qp1l2

2N(λ

(−)0 (R, R)

)−p1V{U(N)}R [K;p]

= (−1)lp1qp1κRV(U(N)R [K;p]. (2.19)

So, to obtain 52 invariant with zero framing, we have to take p1 = −5. For all the non-torus knotsin Fig. 1, we have presented zero framed knot invariants in Appendix A.

Similarly the invariant for link 62 with linking number lk = 3 with framing p1, p2 on thecomponent knots will be

V{U(N)}(R1,R2)

[62;p1,p2] =2∏

i=1

{(−1)lRi

pi qpiκRi

}q

3lR1lR2

N

∑s,t,s′

εR1,R2s

√dimq s

× εR1,R2s′

√dimq s′(λ(+)

s (R1,R2))−3

ats

[R2 R1

R2 R1

]× (λ(−)

t (R1,R2))−2

ats′[

R1 R2

R1 R2

](λ

(+)

s′ (R1, R2))−1

, (2.20)

where lRiis the total number of boxes in the Young diagram representation Ri placed on the

component knots in link 62. The invariants for the non-torus links in Fig. 2 are presented inAppendix A. However to see the polynomial form of these link invariants, we have to determinethe coefficients of the duality matrices. Unlike the SU(2) duality matrices [5,6], we do not havea closed form expression for SU(N) duality matrices.

In the following section, we will use equivalence of states to obtain the sign (2.13), (2.7) andalso derive identities satisfied by the coefficients of the duality matrix. This enables the evaluationof duality matrix elements for some class of representations.

Once we have these coefficients, we could evaluate the framed link invariants and obtain thereformulated invariants [17]


fR1,R2,...,Rr (q, λ) =∞∑

d,m=1

(−1)m−1 μ(d)

dm


r∏α=1

χRα

(C

((m∑

j=1

�k(αj)

)d

))

×m∏

j=1

|C(�k(αj))|�αj ! χRαj

(C(�k(αj)

))×V

(U(N))R1j ,R2j ,...,Rrj

[L; {pα}](qd,λd

), (2.21)

where μ(d) is the Moebius function defined as follows: if d has a prime decomposition ({pi}),d =∏a

i=1 pmi

i , then μ(d) = 0 if any of the mi is greater than one. If all mi = 1, then μ(d) =(−1)a . The second sum in the above equation runs over all vectors �k(αj), with α = 1, . . . , r andj = 1, . . . ,m, such that

∑rα=1 |�k(αj)| > 0 for any j and over representations Rαj . Further �kd

is defined as follows: (�kd)di = ki and has zero entries for the other components. Therefore, if�k = (k1, k2, . . .), then

�kd = (0, . . . ,0, k1,0, . . . ,0, k2,0, . . .), (2.22)

where k1 is in the d-th entry, k2 in the 2d-th entry, and so on. Here C(�k) denotes the conjugacyclass determined by the sequence (k1, k2, . . .) (i.e. there are k1 1-cycles, k2 2-cycles, etc.) inthe permutation group S� (� =∑j jkj ). For a Young tableau representation R with � number

of boxes, χR(C(�k)) gives the character of the conjugacy class C(�k) in the representation R.The explicit relation of the above expression in terms of Chern–Simons invariants are presentedin Appendix D for few representations. The reformulated invariants for r-component links areexpected to obey Labastida–Marino–Vafa conjecture [8]

f(R1,R2,...,Rr )(q, λ) = (q1/2 − q−1/2)r−2∑Q,s

N(R1,...,Rr ),Q,sqsλQ, (2.23)

where N(R1,...,Rr ),Q,s are integers. After determining the identities and some of the matrix el-ements of the duality matrices in the following two sections, we will obtain the polynomialinvariants of the non-torus knots and links in Appendix B and Appendix C.

3. Duality matrix identities

We had elaborated in the previous section on writing states (2.7), (2.8), (2.9) correspondingto Chern–Simons functional integral on three-balls. We can determine the following coefficientsof the duality matrix by comparing Eq. (2.15) and Eqs. (2.7), (2.9):

a0s

[R1 R1R2 R2

]= ε(R1,R2)

s

√dimq s√

dimq R1 dimq R2. (3.1)

This relation along with the property (2.17) suggests that

ats

[R1 R2R3 R4

]= εR1εR2εR3

εR4

√dimq s

√dimq t

{R1 R2 t

R3 R4 s

}, (3.2)

where εRi= ±1 = εRi

and ε0 = 1. The term in parenthesis is similar to the SU(2) quantumWigner 6j symbol but requires appropriate conjugation of representations under interchange ofcolumns in the following way:


Fig. 5. Unknot drawn in two equivalent ways.{R1 R2 t

R3 R4 s

}={

t R2 R1s R4 R3

}={

R1 t R2R3 s R4

}. (3.3)

Using (2.17) and the relation to quantum Wigner symbol with the above properties, the SU(N)

duality matrix can be called as SU(N) quantum Racah coefficients and hence we propose thatthe coefficients obey the following property:

ats

[R1 R2R3 R4

]=

√dimq t dimq s√

dimq R1 dimq R3εR1εR3

(εsεt )−1aR1R3

[t R2s R4

]. (3.4)

Using this property, we can relate the sign in Eq. (3.1) as

ε(R1,R2)s = εR1εR2(εs)

−1. (3.5)

Starting from the state ν(1),(2) in Eq. (2.8) and the duality relation (2.15), we observe that theRacah coefficients must obey the following identities:∑

s

ats

[R1 R2R3 R4

]at ′s

[R1 R2R3 R4

]= δtt ′ , (3.6)

∑t

ats

[R1 R2R3 R4

]ats′[

R1 R2R3 R4

]= δss′ . (3.7)

3.1. Fixing signs of the braiding eigenvalues

From Fig. 5, we can write the invariant for the unknot in two equivalent ways giving thefollowing constraint equation:∑

s

dimq sλ(+)s (R,R) = λ

(−)0 (R, R)dimq R. (3.8)

Taking ε(−)

0;R,R= 1, we can determine the signs ε

(+)s;R,R

which satisfies the above equation. We can

write a general form for the sign for a class of representations Rn =� �n

, the irreduciblerepresentations ρ� ∈ Rn ⊗ Rn in SU(N)k Wess–Zumino conformal field theory in the large k

limit will be� �n

⊗Rn

� �n

Rn

=n⊕

�=0ρ�

� �n−� � �2�


The sign ε(+)ρ�;Rn,Rn

= (−1)(n−�). Similarly, for antiparallelly oriented strands, the irreducible rep-

resentations in ρ� ∈ Rn ⊗ Rn are

. . . . . .� �n

⊗Rn

� �n

Rn

=n⊕

�=0

. . . .

ρ�

� ��

Here boxes with dot represents a column of length N − 1. We take the sign ε(−)

ρ�;Rn,Rn= (−1)�

which is +1 for the singlet � = 0. We can generalise these results for tensor product of twodifferent symmetric representations ρ� ∈ Rn ⊗ Rm and ρ� ∈ Rn ⊗ Rm as

ε(+)ρ�

(Rn,Rm) = (−1)n+m

2 −�, ε(−)

ρ�(Rn, Rm) = (−1)

n−m2 −�, (3.9)

where we assume n � m and � = (n − m)/2, (n − m)/2 + 1, . . . , (n + m)/2. Similarly forantisymmetric representations Rn placed on antiparallelly oriented strands, the irreducible rep-

resentations ˜ρ� ∈ Rn ⊗ ¯Rn are

�

�

n ⊗Rn

�

�

N−n¯Rn =

n⊕�=0

˜ρ�

�

�

�

�

�N−2�

For this case, the sign ε(−)

˜ρ�;Rn,¯Rn

= (−1)�. If we replace N −n = n in the above tensor product,

we get irreducible representations ρ� ∈ Rn ⊗ Rn for parallelly oriented strands carrying antisym-metric representation whose sign will be ε

(+)

ρ�;Rn,Rn= (−1)2n−�. The signs for antisymmetric

representations can be similarly generalised for ρ� ∈ Rn ⊗ Rm and ˜ρ� ∈ Rn ⊗ ¯Rm as

ε(+)

ρ�;Rn,Rm= (−1)n+m−�, ε

(−)

˜ρ�;Rn,¯Rm

= (−1)n−m−�, (3.10)

where n � m&n + m � N and � = 0,1, . . . ,m for parallel strands. Similarly for antiparallelstrands with N − m� n, � = n − m,n − m + 1, . . . , n.

It is possible to fix the signs for the mixed representation but cannot be written in the mostgeneral form as done for the symmetric and the antisymmetric representations. Some of themixed representation signs are given in the earlier papers [9,18]. For simplicity, we will confineto the symmetric or the antisymmetric representations placed on the component knots with thedefined signs which will be useful for writing the Racah coefficients. In the following subsection,we will study equivalence of states which is needed to obtain the Racah coefficients.

3.2. More identities of Racah coefficients from equivalence of states

We can view the two three-balls in Fig. 6 as two equivalent states: The three-ball correspond-

ing to the state |Ψ0(1)〉 can be glued with a similar three-ball with oppositely oriented S2 boundary

to give two unknots. So, this state can be represented as


Fig. 6. Two equivalent three-balls.

∣∣Ψ0(1)⟩= ε(R1,R2)

√dimq R1 dimq R2

∣∣φ0(R2R1R1R2)(1)⟩

(3.11)

= (b(+)2

)−1b

(−)3 ν(1),(2)

∣∣χ(2)0

⟩=∑

s

a0s

[R2 R2R1 R1

](λ(+)

s (R1,R2))−1

ats

[R2 R1R2 R1

]× λ

(−)t (R2, R1)

∣∣φt (R2R1R1R2)(1)⟩. (3.12)

From Eqs. (3.11), (3.12) and similar relations for braid word B = b(+)2 (b

(−)3 )−1, we can deduce

the following identity∑s

a0s

[R2 R2R1 R1

](λ(+)

s (R1,R2))∓1

ats

[R2 R1R2 R1

]= ε(R1,R2)

(λ

(−)t (R2, R1)

)∓1at0

[R2 R1R1 R2

]. (3.13)

Using the data from SU(2) [6], we can fix the signs ε(R1,R2) and the signs in the dualitymatrix for the class of symmetric or antisymmetric representations. Suppose we take sym-metric representations for R1, R2 then the sign ε(Rn,Rm) = (−1)min(n,m) and the signs inthe duality matrix is εRn = (−1)n/2. Similarly, for antisymmetric representations for R1, R2,ε(Rn, Rm) = (−1)min(2n,2m) and ε

Rn= (−1)n. With this sign convention, the above identity en-

ables fixing some of the coefficients of the duality matrix which we tabulate in the next section.The well-known braiding identity relates the two three-balls with two S2 boundaries as picto-

rially shown in Fig. 7. Operating these braiding operators on the two S2 boundary states, we canobtain the following identity for the duality matrix:


Fig. 7. Braiding identity.

Fig. 8. Six-punctured boundary.

∑s,t

λ(+)s (R1,R2)ast

[R2 R1R3 R4

]{λ

(−)t (R1, R3)

}−1ast

[R2 R3R1 R4

]{λ

(−)

s(R2, R3)

}−1

=∑

s,t,s′,t ′ast

[R1 R2R3 R4

]{λ

(−)t (R2, R3)

}−1as′t

[R1 R3R2 R4

]{λ

(−)

s′ (R1, R3)}−1

× as′t ′[

R3 R1R2 R4

]λ

(+)

t ′ (R1,R2)ast ′[

R3 R2R1 R4

]. (3.14)

3.2.1. Six punctured S2 boundariesWe have obtained these identities by studying equivalence of three-balls with four-punctured

S2 boundaries. We shall now look at a generalisation of Fig. 6 for three-balls with six-puncturedS2 boundaries as depicted in Fig. 8 where the braiding operator B = g6 = ({b(+)

4 }−1b(−)3 ) ×

({b(+)5 }−1b

(−)4 {b(+)

3 }−1b(−)2 ).

The Chern–Simons functional integral on these three-balls corresponds to a state in a space ofsix-point correlator conformal blocks in SU(N)k Wess–Zumino conformal field theory. They canbe expanded in a convenient six-point conformal block bases. Two such basis states are drawn


Fig. 9. Six-point conformal block bases.

in Fig. 9. In terms of these six-point conformal block bases, we can relate the state |Ψ 〉 and thestate |ξ 〉 as follows:

|Ψ 〉 = ε(R1,R2,R3)

3∏i=1

√dimq Ri

∣∣φ0,0,0(R1,R2,R3, R3, R2, R1)⟩= Bν(1),(2)|ξ 〉

= B3∏

i=1

√dimq Ri

∣∣φ0,0,0(R1, R1,R2, R2,R3, R3)⟩, (3.15)

where ε(R1,R2,R3) = ±1 = ε(R1,R2)ε(R1,R3)ε(R2,R3) for symmetric and antisymmetricrepresentations. Applying the braiding operator B = g6 on the six-point conformal block in theabove equation, we obtain the following relation:

ε(R1,R2,R3)∣∣φ0,0,0(R1,R2,R3, R3, R2, R1)

⟩=∑

λ(−)q1

(R1,R2){λ(+)

s1(R1, R2)

}−1λ(−)

μ1(R1,R3)

{λ(+)

ν1(R1, R3)

}−1

× λ(−)p2

(R2,R3){λ(+)

μ2(R2, R3)

}−1at10

[0 R2R2 0

]a0q1

[R1 R1R2 t1

]at1s1

[p1 R1R2 0

]× ap1q1

[R1 R2R1 t1

]at2s1

[p1 R2R1 0

]a0μ1

[t2 R1R3 R3

]at2p2

[p1 R2R3 ν1

]× aν1μ1

[t2 R3R1 R3

]at3p2

[p1 R3R2 ν1

]aν1μ2

[t3 R2R3 R1

]aν2μ2

[t3 R3R2 R1

]× at3p3

[p1 R3R3 ν2

] ∣∣φp1,p3,ν2(R1,R2,R3, R3, R2, R1)⟩. (3.16)

We can simplify the RHS of the above expression using the property (2.17). Further, the sum-mation over index q1, μ1 can be done using the identity (3.13). The simplified equation suggestsanother identity for the Racah coefficients:∑

s

ats

[R1 R2R3 R4

]εsq

±Cs2 als

[R1 R3R2 R4

]

= (εt εl)−1q

∓(Ct +Cl )

2

4∏i=1

(εRi

q±(CRi

)

2)atl

[R2 R1R3 R4

]. (3.17)


This is a generalisation of identity (3.13). Using this identity, we can do the summation overindex p2 and μ2 and further simplify the RHS of the expression (3.16). The close similarity ofthese SU(N) coefficients to the SU(2) Racah coefficient identities suggests that all the identitiesof SU(2) quantum coefficients must be generalisable to the SU(N) coefficients and hence wepostulate the following identity

∑l1

ar2l1

[r1 R3R4 R5

]ar1l2

[R1 R2l1 R5

]al1l3

[l2 R2R3 R4

]

= ar2l2

[R1 l3R4 R5

]ar1l3

[R1 R2R3 r2

], (3.18)

where we have appropriately chosen conjugate representations which are consistent with the def-inition of the Racah matrix. Again, using the above identity with l3 = 0, we can do the summationover ν1 index in the simplified RHS of Eq. (3.16). Finally, we see the Eq. (3.16) reducing to

∣∣φ0,0,0(R1,R2,R3, R3, R2, R1)⟩= ap10

[R1 R2R2 R1

] ∣∣φp1,0,p1(R1,R2,R3, R3, R2, R1)⟩.

(3.19)

Using the properties of the duality matrix, RHS can be seen to be LHS. This elaborate exer-cise on the equivalence of two states corresponding to six-punctured boundary confirms that thecorrectness of the identity (3.18). Armed with these identities, we try to determine the Racahcoefficients for some representations which we present in the next section.

4. SU(N) quantum Racah coefficients

We shall use the properties and identities derived in the previous sections to obtain the dualitymatrix coefficients which will be useful for computing the non-torus knot and non-torus links.

For knots, all the strands carry same representation. So for obtaining non-torus knot invariants,we have to evaluate two types of Racah matrices, namely,

als

[R R

R R

], als

[R R

R R

], (4.1)

where first type can be shown to be a symmetric matrix from the properties of the Racah matrix.

We could evaluate the symmetric first type Racah coefficients for R = , , using Eqs. (3.6),(3.7) [2,3]. However, for the second type Racah matrix we could only evaluate the coefficientsfor the fundamental representation (R = ).

Similarly for the two-component links, we can place two representations R1, R2 on the com-ponent knots. In this case, we can have three types of Racah matrices as follows:

alt

[R1 R1R2 R2

], als

[R1 R2R2 R1

], alt

[R1 R2R1 R2

]. (4.2)

Now, we will present the Racah coefficients for some representations which will be useful tocompute the non-torus knot and link polynomials in U(N) Chern–Simons theory.


4.1. Coefficients when R1 is fundamental

1. For the simplest fundamental representation R = , the two types of Racah coefficientmatrices are [2,3]:

ats

[R = R = .

R = R = .

]= 1

dimq R

⎛⎝ s = 0 R = .

t = 0 −1√[N − 1][N + 1]

.√[N − 1][N + 1] 1

⎞⎠ ,

ats

[R = R =R = . R = .

]= 1

dimq R

⎛⎜⎜⎜⎜⎝s = 0 .

t = −√

[N ][N−1][2]

√[N ][N+1]

[2]√[N ][N+1]

[2]√

[N ][N−1][2]

⎞⎟⎟⎟⎟⎠ ,

where dimq(R = ) = [N ].2. Next, we look at Racah coefficient matrices where R1 = �= R2. This will be useful for the

computation of two-component links.

ats

[R1 = R = .

R2 = R2 = . .

]= 1

K

⎛⎜⎜⎜⎝s = .

t = 0 −√[N ]√

[N−1][N ][N+2][2]

.√

[N−1][N ][N+2][2]

√[N ]

⎞⎟⎟⎟⎠ ,

where K = √dimq R1 dimq R2. Similarly, the second and third type Racah matrix coefficientsfor R1 = , R2 = are

ats

[R1 = R2 =R1 = . R2 = . .

]=

⎛⎜⎜⎜⎜⎝s = .

t = −√

[N−1][3][N+1]

√[2][N+2][3][N+1]√

[2][N+2][3][N+1]

√[N−1]

[3][N+1]

⎞⎟⎟⎟⎟⎠ ,

ats

[R1 = R2 =R2 = . . R1 = .

]= 1

K

⎛⎜⎜⎜⎜⎝s = 0 .

t = −√

[N−1][N ][N+1][3]

√[N ][N+1][N+2]

[2][3]√[N ][N+1][N+2]

[2][3]√

[N−1][N ][N+1][3]

⎞⎟⎟⎟⎟⎠ .

3. Interestingly, we could find the coefficients for R1 = , R2 = n� �using the identi-

ties. The corresponding conjugate representations are R1 = . , R2 = . . . . .n� �. The three types of

Racah coefficients for this class of representations are

ats

[R1 R1

R2 R2

]= 1

K

⎛⎜⎜⎜⎜⎝s = n−1� �

. n+1� �

t = 0 −√

[N ][N+1]···[N+n−2][n−1]!

√[N−1][N ]···[N+n]

[N+n−1][n]!.

√[N−1][N ]···[N+n]

[N+n−1][n]!√

[N ][N+1]···[N+n−2][n−1]!

⎞⎟⎟⎟⎟⎠ ,


ats

[R1 R2

R1 R2

]=

⎛⎜⎜⎜⎜⎝s =

n� �n+1� �

t = n−1� � −√

[N−1][n+1][N+n−1]

√[n][N+n]

[n+1][N+n−1]. n+1� � √

[n][N+n][n+1][N+n−1]

√[N−1]

[4][N+2]

⎞⎟⎟⎟⎟⎠ ,

ats

[R1 R2

R2 R1

]= 1

K

⎛⎜⎜⎜⎜⎜⎝s = 0 .

t =n� �

−√

[N−1][N ]···[N+n−1][n+1][n−1]!

√[N ][N+1]···[N+n]

[n+1]!n+1� � √

[N ][N+1]···[N+n][n+1]!

√[N−1][N ]···[N+n−1]

[n+1][n−1]!

⎞⎟⎟⎟⎟⎟⎠ ,

where K =√dimq R1 dimq R2.

4. Similar exercise could be done for R1 = , R2 = , and their conjugate representations

are R1 = . , R2 = N−2��

. These representations give

ats

[R1 R1

R2 R2

]= 1√

dimq R1 dimq R2

⎛⎜⎜⎜⎝s = .

t = 0√[N ]

√[N−2][N ][N+1]

[2].

√[N−2][N ][N+1]

[2] −√[N ]

⎞⎟⎟⎟⎠ ,

ats

[R1 R2

R1 R2

]=

⎛⎜⎜⎜⎜⎝s = .

t =√

[N+1][3][N−1] −

√[2][N−2][3][N−1]

−√

[2][N−2][3][N−1] −

√[N+1]

[3][N−1]

⎞⎟⎟⎟⎟⎠ ,

ats

[R1 R2

R2 R1

]= 1√

dimq R1 dimq R2

⎛⎜⎜⎜⎜⎜⎝s = 0 .

t =√

[N−1][N ][N+1][3] −

√[N−2][N ][N−1]

[2][3]

−√

[N−2][N ][N−1][2][3] −

√[N−1][N ][N+1]

[3]

⎞⎟⎟⎟⎟⎟⎠ .

5. These results can be generalised for R1 = and R2 = n��, whose conjugate representations

are R1 = . , R2 = N−n��

as follows:

ats

[R1 R1

R2 R2

]= 1

K

⎛⎜⎜⎜⎜⎜⎝s = �n−1

�

.n��

t = 0√

[N ][N−1]···[N−n+2][n−1]!

√[N+1][N ]···[N−n]

[N−n+1][n]!.

√[N+1][N ]···[N−n]

[N−n+1][n]! −√

[N ][N−1]···[N−n+2][n−1]!

⎞⎟⎟⎟⎟⎟⎠ ,


ats

[R1 R2

R1 R2

]=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

s = �n−1�

.n��

t = �n−1�

√[N+1]

[N+n][N−n+1] −√

[n][N−n][n+1][N−n+1]

n+1��

−√

[n][N−n][n+1][N−n+1] −

√[n][N−n]

[n+1][N−n+1]

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,

ats

[R1 R2

R2 R1

]= (−1)n

K

⎛⎜⎜⎜⎜⎜⎝s = 0 .

t = �n−1�

√[N+1][N ]···[N−n+1]

[n+1][n−1]! −√

[N ][N−1]···[N−n][n]!

n+1��

−√

[N ][N−1]···[N−n][n]! −

√[N+1][N ]···[N−n+1]

[n+1][n−1]!

⎞⎟⎟⎟⎟⎟⎠ ,

where K =√dimq R1 dimq R2.

4.2. Coefficients for R1 symmetric or antisymmetric representations

1. For R = , R = . .

ats

[R R

R R

]= 1

K

⎛⎜⎜⎜⎜⎜⎜⎜⎝

s =ρ00

ρ1.ρ2. .

t = ρ0√

dimq ρ0 −√dimq ρ1√

dimq ρ2

ρ1 −√dimq ρ1dimq ρ2

dimq R−1 − 1√

dimq ρ1 dimq ρ2

dimq R−1

ρ2√

dimq ρ2

√dimq ρ1 dimq ρ2

dimq R−1dimq ρ1

dimq R−1 − 1

⎞⎟⎟⎟⎟⎟⎟⎟⎠,

where

K = dimq R = [N ][N + 1][2] , dimq ρ0 = 1, dimq ρ1 = [N + 1][N − 1],

dimq ρ2 = [N − 1][N ]2[N + 3][2]2

. (4.3)

The second type Racah matrix coefficients are

ats

[R R

R R

]

= 1

K

⎛⎜⎜⎜⎜⎜⎜⎜⎝

t s =ρ00

ρ1.ρ2. .

(ρ1)√

dimq ρ1 −[N][N+1][2]

√1

[3][N][2]√

[N+3][N+1][3]

(ρ2) −√dimq ρ2 x

√[N][N+2]

[4][2] y

√[N][N+1][N+2][N+3]

[4][2](ρ3)

√dimq ρ3 u

√[N−1][N][N+2][N+3]

[4][3][2] v

√[N−1][N][N+1][N+2]

[4][3][2]

⎞⎟⎟⎟⎟⎟⎟⎟⎠,

where the quantum dimensions of the representations are


dimq ρ1 = [N − 1][N ]2[N + 1][2]2[3] , dimq ρ2 = [N − 1][N ][N + 1][N + 2]

[4][2] ,

dimq ρ3 = [N ][N + 1][N + 2][N + 3][4][3][2] . (4.4)

The variables x, y, u, v are

x = [N + 3][N ][N + 2] − [N − 1], y = [N ]

[N + 2] ,

u = [2][N + 1][N + 2] , v = [N + 1][2]

[N + 2] − 1.

2. For R = , R = N−2��

ats

[R R

R R

]= 1

K

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝s =ρ0

0ρ1.

N−2��

ρ2

t = ρ0√

dimq ρ0 −√dimq ρ1√

dimq ρ2

ρ1 −√dimq ρ1dimq ρ2

dimq R−1 − 1√

dimq ρ1 dimq ρ2

dimq R−1

ρ2√

dimq ρ2

√dimq ρ1 dimq ρ2

dimq R−1dimq ρ1

dimq R−1 − 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,

where

K = dimq R = [N ][N − 1][2] , dimq ρ0 = 1, dimq ρ1 = [N + 1][N − 1],

dimq ρ2 = [N + 1][N ]2[N − 3][2]2

. (4.5)

The second type Racah matrix coefficients are

ats

[R R

R R

]

= 1

K

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

t s =ρ00

ρ1.

N−2��

ρ2

(ρ1)√

dimq ρ1[N][N−1]

[2]√

1[3]

[N][2]√

[N−3][N−1][3]

(ρ2) −√dimq ρ2 x

√[N][N−2]

[4][2] y

√[N][N−1][N−2][N−3]

[4][2]

(ρ3)√

dimq ρ3 −u

√[N+1][N][N−2][N−3]

[4][3][2] v

√[N+1][N][N−1][N−2]

[4][3][2]

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

where

dimq ρ1 = [N − 1][N ]2[N + 1][2]2[3] ,

dimq ρ2 = [N + 1][N ][N − 1][N − 2][4][2] ,

dimq ρ3 = [N ][N − 1][N − 2][N − 3][4][3][2] ,


and the variables are

x = [N + 1] − [N − 3][N ][N − 2] , y = [N ]

[N − 2] ,

u = [2][N − 1][N − 2] , v = [N − 1][2]

[N − 2] − 1. (4.6)

3. For R1 = �= R2 = , which is will be useful for the computation of links, the secondtype Racah coefficients are

ats

[R1 R2

R1 R2

]

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

t s =ρ00

ρ1.

ρ1. .

ρ0 z1

√ [N−1][N][N+1][N+2][4][3][2] − [2]

[N+1]√ [N][N+1]

[4][3][2]

[N+2]√ [N+2][N+4]

[4][2]ρ1 −z1[2]

√ [N−1][N+1][N+2][N+3][2][3][5] z2

√ [N+1][N+3][3][5] z3

√ [N][N+2][N+3][N+4][2][5]

ρ2 z1[3]√ [N+1][N+2][N+3][N+4]

[5][4][3][2] z4[3]√ [N−1][N+1][N+3][N+4]

[5][4][3] z3

√ [N−1][N][N+2][N+3][5][4][2]

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,

where ρ0 = , ρ1 = , ρ2 = and the variables zi ’s are

z1 = [2][N + 1][N + 2] , z2 = z1

( [N + 3][N + 4] − [N ][N−1][3]

[N ][3] + [N + 4]

),

z3 = [2][N + 2][N + 3] , z4 = [2]

[N + 1][N + 3] .The Racah coefficients of third type is given by

ats

[R1 R2

R2 R1

]

= 1√K

⎛⎜⎜⎜⎜⎜⎜⎜⎝t s = 0

ρ1.

ρ2. .

ρ0√

dimq ρ0 − [N][N+1][3]

√[N+3]

[4][N][3]√

[N+1][N+3][N+4][2][4]

ρ1 −√dimq ρ1 −([N − 2] + [N ] − [N + 4]) [N+1][2][3]

√ [N][5]

[2][N][3]

√[N][N+1][N+4]

[2][5]ρ1

√dimq ρ2 [N + 1]

√[N−1][N][N+4]

[4][5] [N ]√

[N−1][N][N+1][2][4][5]

⎞⎟⎟⎟⎟⎟⎟⎟⎠ .

(4.7)

The quantum dimensions of the representations in terms of the q-numbers are

√K =√dimq R1 dimq R2 = [N ][N + 1]

[2]

√[N + 2]

[3] ,

dimq ρ0 = [N − 1][N ]2[N + 1][N + 2][4][3][2] ,

dimq ρ1 = [N − 1][N ][N + 1][N + 2][N + 3][2][3][5] ,

dimq ρ2 = [N ][N + 1][N + 2][N + 3][N + 4][2][3][4][5] .

Using the identities, it should be possible to generalise these Racah matrices for R2 =� �n

, which are again 3 × 3 matrices.


4. Equivalently, we could write the Racah matrix coefficients when R1 = �= R2 where R2 is

totally antisymmetric n-th rank tensor (represented by n-vertical box). For R1 = and R2 = ,the second type Racah coefficient matrix is

ats

[R1 R2

R1 R2

]

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝t s =

ρ0

ρ1. N−4�

�

ρ2

ρ0 z1

√[N+1][N ][N−1][N−2]

[4][3][2][2]

[N−1]√

[N ][N−1][4][3]

[2][N−2]

√[N−2][N−4]

[4][2]ρ1 −z1[2]

√[N+1][N−1][N−2][N−3]

[2][3][5] z2

√[N−1][N−3]

[3][5] z3

√[N ][N−2][N−3][N−4]

[2][5]ρ2 z1[3]

√[N−1][N−2][N−3][N−4]

[5][4][3][2] −z4[3]√

[N+1][N−1][N−3][N−4][5][4][3] z3

√[N+1][N ][N−2][N−3]

[5][4][2]

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

where ρ0 = , ρ1 = , ρ2 = and the variables are

z1 = [2][N − 1][N − 2] , z2 = z1

(−[N − 3][N − 4] + [N ][N+1][3]

[N ][3] + [N − 4]

),

z3 = [2][N − 2][N − 3] , z4 = [2]

[N − 1][N − 3] .Finally, with these data available, we evaluate the polynomial form for the non-torus knots andlinks in Figs. 1 and 2 in Appendix B and Appendix C using the formula in Appendix A. Fromthese invariants, the reformulated invariants fR[K], fR1,R2[L] (2.21) are obtained and shown toobey Eq. (2.23).

5. Discussion and conclusion

In this paper, we have attempted a challenging problem of obtaining matrix elements of theduality matrix which has properties and identities similar to the quantum Racah coefficients.Particularly, we derived these identities and properties by studying the equivalence of states inthe space of correlator conformal blocks in the SU(N)k Wess–Zumino conformal field theory.

We have tabulated the Racah coefficients for some class of representations which will beuseful to compute non-torus knots and non-torus two component links. We have presented thepolynomial form for all the non-torus knots and non-torus links in Figs. 1 and 2 (see Appendix Band Appendix C) and obtained their reformulated invariants. These invariants obey the conjec-tured form (2.23) [7,8] confirming the correctness of our Racah coefficients in Section 4.

We believe that there must be a systematic way of writing a closed form expression similarto the expression obtained for SU(2) quantum Racah coefficients [5,6]. There are papers in theliterature addressing classical Racah and quantum Racah coefficients. Unfortunately, we do notsee such explicit coefficients in Section 4 to compare. We hope to study those papers which mayhelp us to obtain a closed form expression for SU(N) quantum Racah coefficients.

There are interesting recent developments relating torus knots to spectral curve in the B-model topological strings [11], Poincare polynomial computation from refined Chern–Simonstheory, Khovanov homology, fivebranes [10,12], and the polynomial invariants from counting ofsolutions in four-dimensional theories [13] We hope to extend these recent works to non-toruslinks and report in future.


Acknowledgements

P.R. would like to thank the hospitality of Center for Quantum spacetime, Sogang University,where this work was completed. This work was supported by the National Research Foundationof Korea (NRF) grant funded by the Korea government (MEST) through the Center for QuantumSpacetime (CQUeST) of Sogang University with grant number 2005-0049409.

Appendix A. Formulae for U(N) link invariants in terms of SU(N) quantum Racahcoefficients and braiding eigenvalues

In this appendix we give the expression of U(N) link invariant for all the non-torus knots andlinks in Fig. 1 and Fig. 2 in terms of SU(N) quantum Racah coefficients and braiding eigenvalues.

A.1. Non-torus knots

V{U(N)}R [41;0] =

∑s,t,s′

εR,Rs

√dimq sε

R,Rs′√

dimq s′(λ(−)s (R,R)

)2ats

[R R

R R

]

× (λ(−)t (R,R)

)−1ats′[

R R

R R

](λ

(+)

s′ (R,R))−1

. (A.1)

V{U(N)}R [52;0] = q(−5κR+ 5l2

2N)∑s,t,s′

εR,Rs

√dimq sε

R,Rs′√


)−2

× ats

[R R

R R

](λ

(−)t (R,R)

)−2ats′[

R R

R R

](λ

(+)

s′ (R,R))−1

. (A.2)

V{U(N)}R [61;0] = q(−2κR+ l2

N)∑s,t,s′

εR,Rs

√dimq sε

R,Rs′√


)2ats

[R R

R R

]

× (λ(−)t (R,R)

)−3ats′[

R R

R R

](λ

(+)

s′ (R,R))−1

. (A.3)

V{U(N)}R [62;0] = q(−2κR+ l2

N)∑

s,t,s′,u,v

εR,Rs

√dimq sεR,R

v

√dimq vλ(+)

s (R,R)ats

[R R

R R

]

× λ(−)t (R, R)ats′

[R R

R R

](λ

(−)

s′ (R, R))−1

aus′[

R R

R R

]× (λ(+)

u (R,R))−2

auv

[R R

R R

](λ(−)

v (R,R))−1

. (A.4)

V{U(N)}R [63;0] =

∑s,t,s′,u,v

εR,Rs

√dimq sεR,R

v

√dimq v

(λ(+)

s (R,R))−2

ats

[R R

R R

]

× (λ(−)t (R, R)

)−1ats′[

R R

R R

]λ

(−)

s′ (R, R)aus′[

R R

R R

]× (λ(+)

u (R, R))auv

[R R

R R

](λ(−)

v (R,R)). (A.5)


V{U(N)}R [72;0] = q(−7κR+ 7l2

2N)∑s,t,s′

εR,Rs

√dimq sε

R,Rs′√


)−2

× ats

[R R

R R

](λ

(−)t (R, R)

)−4ats′[

R R

R R

](λ

(+)

s′ (R,R))−1

. (A.6)

V{U(N)}R [73;0] = q(7κR− 7l2

2N)∑s,t,s′

εR,Rs

√dimq sε

R,Rs′√

dimq s′(λ(−)s (R, R)

)3ats

[R R

R R

]

× (λ(+)t (R, R)

)3ats′[

R R

R R

]λ

(−)

s′ (R, R). (A.7)

V{U(N)}R [74;0] = q(7κR− 7l2

2N)∑

s,t,s′,u,v

εR,Rs

√dimq sεR,R

v

√dimq vλ(+)

s (R,R)ats

[R R

R R

]

× (λ(−)t (R, R)

)2ats′[

R R

R R

]λ

(+)

s′ (R, R)aus′[

R R

R R

]× (λ(−)

u (R, R))2

auv

[R R

R R

]λ(+)

v (R,R). (A.8)

V{U(N)}R [75;0] = q(−7κR+ 7l2

2N)∑

s,t,s′,u,v

εR,Rs

√dimq sεR,R

v

√dimq v

(λ(−)

s (R, R))−1

× ats

[R R

R R

](λ

(+)

t(R,R)

)−1ats′[

R R

R R

](λ

(−)

s′ (R,R))−2

× aus′[

R R

R R

](λ

(+)u (R,R)

)−2auv

[R R

R R

](λ(−)

v (R,R))−1

. (A.9)

V{U(N)}R [76;0] = q(−3κR+ 3l2

2N)∑

s,t,s′,u,v

εR,Rs

√dimq sεR,R

v

√dimq v

(λ(−)

s (R, R))−2

× ats

[R R

R R

](λ

(−)t (R,R)

)2ats′[

R R

R R

](λ

(−)

s′ (R, R))−1

× aus′[

R R

R R

](λ(+)

u (R, R))−1

auv

[R R

R R

](λ(−)

v (R,R))−1

. (A.10)

V{U(N)}R [77;0] = q(κR− l2

2N)

∑s,t,s′,u,v,w,x

εR,Rs

√dimq sεR,R

x

√dimq x

(λ(+)

s (R,R))

× ats

[R R

R R

](λ

(−)t (R,R)

)ats′[

R R

R R

](λ

(−)

s′ (R,R))−1

× aus′[

R R

R R

](λ

(+)u (R, R)

)−1auv

[R R

R R

](λ(−)

v (R, R))−1

× awv

[R R

R R

]λ(−)

w (R, R)awx

[R R

R R

]λ(+)

x (R,R). (A.11)


V{U(N)}R [81;0] = q(−4κR+ 2l2

N)∑s,t,s′

εR,Rs

√dimq sε

R,Rs′√


)2ats

[R R

R R

]

× (λ(−)t (R,R)

)−5ats′[

R R

R R

](λ

(+)

s′ (R,R))−1

. (A.12)

V{U(N)}R [92;0] = q(−7κR+ 7l2

2N)∑s,t,s′

εR,Rs

√dimq sε

R,Rs′√


)−2

× ats

[R R

R R

](λ

(−)t (R,R)

)−6ats′[

R R

R R

](λ

(+)

s′ (R,R))−1

. (A.13)

V{U(N)}R [101;0] = q(−6κR+ 3l2

N)∑s,t,s′

εR,Rs

√dimq sε

R,Rs′√


)2× ats

[R R

R R

](λ

(−)t (R,R)

)−7ats′[

R R

R R

](λ

(+)

s′ (R,R))−1

. (A.14)

For framed knots K with framing number p, the invariants will be related to the zero-framedknot invariants as

V{U(N)}R [K;p] = qpκRV

{U(N)}R [K;0]. (A.15)

A.2. Non-torus links

In the context of links, we can place different representations on the component knots. Theinvariants are hence called multicoloured links.

V{U(N)}(R1,R2)

[62;0,0] = q3lR1

lR2N

∑s,t,s′

εR1,R2s

√dimq sε

R1,R2s′

√dimq s′(λ(+)

s (R1,R2))−3

× ats

[R2 R1

R2 R1

](λ

(−)

t(R1,R2)

)−2

× ats′

[R1 R2

R1 R2

](λ

(+)

s′ (R1, R2))−1

. (A.16)

V{U(N)}(R1,R2)

[63;0,0] = q(−2κR2+l2R2N

− 2lR1lR2

N)∑

s,t,s′,u,v

εR1,R2s

√dimq sεR1,R2

v

×√dimq vλ(+)s (R1,R2)

× ats

[R1 R2

R1 R2

]λ

(−)

t(R1, R2)ats′

[R1 R2

R2 R1

](λ

(−)

s′ (R2, R2))−2

× aus′[

R1 R2

R2 R1

]λ(−)

u (R1,R2)auv

[R2 R1

R2 R1

]λ(+)

v (R1, R2).

(A.17)


V{U(N)}(R1,R2)

[71;0,0] = q(−κR2+l2R22N

+ lR1lR2

N)∑

s,t,s′,u,v

εR1,R2s

√dimq sεR1,R2

v

×√dimq v(λ(+)

s (R1,R2))

× ats

[R1 R2

R1 R2

]λ

(−)

t(R1, R2)ats′

[R1 R2

R2 R1

](λ

(−)

s′ (R2, R2))−1

× aus′[

R1 R2R2 R1

]λ(+)

u (R1, R2)−3auv

[R2 R1R2 R1

](λ(−)

v (R1,R2))−1

.

(A.18)

V{U(N)}(R1,R2)

[72;0,0] = q(−κR2+l2R22N

− lR1lR2

N)∑

s,t,s′,u,v

εR1,R2s

√dimq sεR1,R2

v

×√dimq v(λ(+)

s (R1,R2))−2

× ats

[R1 R1

R2 R2

](λ

(−)

t(R2, R2)

)−1ats′[

R1 R1

R2 R2

](λ

(−)

s′ (R1, R2))

× aus′[

R1 R2R1 R2

]λ

(+)u (R1,R2)

2auv

[R2 R1R2 R1

]λ(−)

v (R1,R2).

(A.19)

V{U(N)}(R1,R2)

[73;0,0] = q(−3κR2+3l2

R22N

)∑

s,t,s′,u,v

εR1,R2s

√dimq sεR1,R2

v

√dimq vλ(+)

s (R1,R2)

× ats

[R1 R2

R1 R2

]λ

(−)

t(R1, R2)ats′

[R1 R2

R2 R1

](λ

(−)

s′ (R2, R2))−3

× aus′[

R1 R2R2 R1

](λ(+)

u (R1, R2))−1

auv

[R2 R1R2 R1

]× (λ(−)

v (R1,R2))−1

. (A.20)

Including the framing numbers p1, p2 on the component knots of these two-component toruslinks L, the framed multicoloured invariant will be

V{U(N)}(R1,R2)

[L;p1,p2] = q(p1κR1+p2κR2 )V{U(N)}(R1,R2)

[L;0,0]. (A.21)

Appendix B. Knot polynomials

In this appendix we present the polynomial form of the U(N) link invariant for non-torus

knots in Fig. 1 for representation whose Young tableau diagrams are , and . The polynomialcorresponding to representation is proportional to HOMFLY-PT polynomial P(λ, t)[K] [19,20] up to unknot U normalisation:

P(λ,q)[K] = VU(N)[K;0]V

U(N)[U ]= (q1/2 − q−1/2)

(λ1/2 − λ−1/2)V

U(N)[K;0]. (B.1)

We list them so that we can directly use them in the computation of reformulated invariants inAppendix D.


1. For fundamental representation R = placed on the knot, the U(N) knot polynomials are

VU(N)[41] = (λ − 1)

λ3/2(q − 1)√

q

[−λ − λq2 + (λ2 + λ + 1)q],

VU(N)[52] = 1

(−1 + q)√

qλ7/2

[q − qλ3 + λ

(−1 + λ2)+ q2λ(−1 + λ2)],

VU(N)[61] = 1

λ5/2(q − 1)√

q

[−λ3 + λ + (λ − λ3)q2 + (λ4 + λ3 − λ2 − 1)q],

VU(N)[62] = (−1 + λ)

(−1 + q)q3/2λ5/2

[−λ − q4λ − q2(1 + 2λ) + q(1 + λ + λ2)

+ q3(1 + λ + λ2)],V

U(N)[63] = − (−1 + λ)

(−1 + q)q3/2λ3/2

[−λ − q4λ + q(1 + λ + λ2)+ q3(1 + λ + λ2)

− q2(1 + 3λ + λ2)],V

U(N)[72] = 1

(−1 + q)√

qλ9/2

[λ(−1 + λ3)+ q2λ

(−1 + λ3)− q(−1 − λ2 + λ3 + λ4)],

VU(N)[73] = λ3/2

(−1 + q)q3/2

[−1 + q + λ2 − qλ3 + q4(−1 + λ2)+ q2(−1 − λ + 2λ2)− q3(−1 + λ3)],

VU(N)[74] = (−1 + λ)λ1/2

(−1 + q)q1/2

[(1 + λ)2 + q2(1 + λ)2 − q

(2 + 2λ + 2λ2 + λ3)],

VU(N)[75] = (−1 + λ)

(−1 + q)q3/2λ9/2

[λ(1 + λ) + q4λ(1 + λ) − q(1 + λ)2

− q3(1 + λ)2 + q2(1 + 2λ + 2λ2)],V

U(N)[76] = − (−1 + λ)

(−1 + q)q3/2λ7/2

[λ2 + q4λ2 − qλ

(2 + 2λ + λ2)

− q3λ(2 + 2λ + λ2)+ q2(1 + 2λ + 3λ2 + λ3)],

VU(N)[77] = (−1 + λ)

(−1 + q)q3/2λ3/2

[λ + q4λ − q

(1 + 2λ + 2λ2)− q3(1 + 2λ + 2λ2)

+ q2(2 + 4λ + 2λ2 + λ3)],V

U(N)[81] = 1

(q − 1)√

qλ7/2

[−λ4 + λ + q2(λ − λ4)+ q(λ5 + λ4 − λ2 − 1

)],

VU(N)[91] = 1

(q − 1)√

qλ11/2

[λ(λ4 − 1

)q2 − (λ5 + λ4 − λ2 − 1

)q + λ

(λ4 − 1

)],

VU(N)[101] = 1

(q − 1)√

qλ9/2

[−λ5 + λ + q2(λ − λ5)+ q(λ6 + λ5 − λ2 − 1

)].


2. For symmetric second rank representation R = , the knot polynomials are

VU(N)[41] = (−1 + λ)(−1 + qλ)

(−1 + q)2q2(1 + q)λ3

[(−1 + λ)λ + 3q3λ2 − q6(−1 + λ)λ2

+ q4λ(−1 + λ2)+ q5λ2(−1 − λ + λ2)

− q(−1 + λ + λ2)+ q2(λ − λ3)],

VU(N)[52] = 1

(−1 + q)2q5(1 + q)λ7

[q(−1 + λ)2 − (−1 + λ)2λ + q9λ4(−1 + λ2)

+ q3(−1 + λ)2λ2(1 + λ + λ2)+ q6λ4(−1 − λ + 2λ2)+ q5λ2(−1 + λ − λ2 + λ3)− q8λ3(−1 + λ − λ2 + λ3)+ q4λ

(−1 + λ + λ2 − λ5)+ q7(λ3 − λ6)],V

U(N)[61] = (−1 + λ)(−1 + qλ)

(−1 + q)2q4(1 + q)λ5

[(−1 + λ)λ + q(−1 + λ)2(1 + λ)

+ q2(−1 + λ)2λ(1 + λ) + q7λ4(−3 + λ2)+ 2q6λ3(−1 + λ2)− 2q3λ2(−1 + λ + λ2)+ q5λ2(−1 + 2λ + 4λ2)− q4λ

(1 − λ − 3λ2 + 2λ3 + λ4)+ q8(λ3 − λ5)],

VU(N)[62] = (−1 + λ)(−1 + qλ)

(−1 + q)2q6(1 + q)λ5

[q + (−1 + λ)λ − qλ2 − q9(−4 + λ)λ2

− q12(−1 + λ)λ2 − q2(−1 + λ)2(1 + λ) + q3λ(−3 + 2λ)

− 2q4(−1 + λ2)+ q10λ(−1 + λ2)+ q11λ2(−1 − λ + λ2)

+ q6(−1 − 3λ + 4λ2)+ q8λ(2 − 3λ2 + λ3)+ q7(1 − 2λ − 3λ2 + λ4)

+ q5(−1 + 3λ + 2λ2 − 2λ3 + λ4)],V

U(N)[63] = − (−1 + λ)(−1 + qλ)

(−1 + q)2q5(1 + q)λ3

[−(−1 + λ)λ + q12(−1 + λ)λ2

+ q3(1 + 3λ − 4λ2)+ q(−1 + λ2)+ q9λ2(−4 + 3λ + λ2)

+ q7(−1 + 4λ + λ2 − 4λ3)+ q5λ(−4 + λ + 4λ2 − λ3)

− q4(2 − 3λ − 3λ2 + λ3)+ q10λ(1 − λ − 2λ2 + λ3)

+ q2(1 − 2λ − λ2 + λ3)− q8λ(1 − 3λ − 3λ2 + 2λ3)

+ q11(λ2 − λ4)+ q6(1 + λ − 9λ2 + λ3 + λ4)],V

U(N)[72] = 1

(−1 + q)2q7(1 + q)λ9

[q(−1 + λ)2 − (−1 + λ)2λ − q4(−1 + λ)2λ

+ q3(−1 + λ)2λ2 + q5(−1 + λ)2λ4(1 + λ + λ2)+ q11λ5(−1 + λ3)+ q8λ5(−1 − λ + 2λ3)+ q7λ3(−1 + λ − λ2 + λ4)− q10λ4(−1 + λ − λ2 + λ4)− q6λ3(1 − 2λ + λ2 − 2λ3 + λ4 + λ5)+ q9(λ4 + λ6 − λ7 − λ8)],


VU(N)[73] = (−1 + λ)λ3(−1 + qλ)

(−1 + q)2q3(1 + q)

[1 + q(−1 + λ) − q12(−1 + λ) + λ

− q14(−1 + λ)λ2 + q13(−1 + λ)2λ(1 + λ) − q2(1 + 2λ)

− q4λ(−4 + λ2)− 2q3(−1 + λ2)+ q5(−2 − λ + 3λ2)

+ q8(−1 + λ + 4λ2 − 2λ3)+ q9(1 − 2λ + λ3)+ q6(1 − 3λ − 2λ2 + λ3)+ q10λ

(2 − 2λ − λ2 + λ3)

+ q7(1 + 3λ − 3λ2 − 2λ3 + λ4)+ q11(−1 + 2λ2 − 2λ3 + λ4)],V

U(N)[74] = (−1 + λ)λ(−1 + qλ)

(−1 + q)2q(1 + q)

[−q10(−1 + λ)λ4 + (1 + λ)2 + 2q(−1 + λ2)

+ q2(−1 − 6λ − λ2 + λ3)+ q9λ3(2 − 2λ − λ2 + λ3)− q3(−4 − 2λ + 5λ2 + λ3)+ q8λ2(3 − 2λ − 3λ2 + 2λ3)+ q5(−2 − 4λ + 6λ2 + 3λ3 − 3λ4)+ q7λ

(2 − 2λ − 4λ2 + 3λ3 + λ4)

− q4(1 − 6λ − 4λ2 + 4λ3 + λ4)+ q6(1 − 2λ − 4λ2 + 5λ3 + λ4 − λ5)],

VU(N)[75] = (−1 + λ)(−1 + qλ)

(−1 + q)2q9(1 + q)λ9

[(−1 + λ)λ − q13(−1 + λ)λ3

+ q2(−1 + λ)3(1 + λ) + q14λ3(1 + λ) − q12λ3(3 + λ)

+ 3q11λ2(−1 + λ2)+ q10λ(−1 + 6λ2)− q3λ

(3 − 4λ + λ3)

+ q(1 − 2λ2 + λ3)+ q8λ

(3 − 5λ − 5λ2 + 3λ3)

+ q9(λ + 5λ2 − 3λ3 − 3λ4)+ q4(2 − λ − 5λ2 + 3λ3 + λ4)+ q7(1 − 3λ − 5λ2 + 6λ3 + λ4)− q6(1 + 2λ − 7λ2 + 2λ4)+ q5(−1 + 5λ − λ2 − 5λ3 + 2λ4)],

VU(N)[76] = (−1 + λ)(−1 + qλ)

(−1 + q)2q6(1 + q)λ7

[(−1 + λ)2λ2 − q12(−1 + λ)λ4

+ q11λ4(−2 + λ2)+ qλ(−2 + 3λ + λ2 − 2λ3)

− q10λ3(2 − 3λ2 + λ3)− q9λ3(−1 − 7λ + 2λ2 + λ3)+ q7λ2(3 − 7λ − 7λ2 + 4λ3)+ q6λ2(−4 − 6λ + 10λ2 + λ3 − λ4)+ q4λ

(−1 + 8λ − λ2 − 7λ3 + λ4)+ q3λ(2 − 8λ2 + 4λ3 + λ4)

+ q8λ2(1 + 5λ − 4λ2 − 4λ3 + 2λ4)+ q5λ

(−2 − 2λ + 10λ2 + 2λ3 − 4λ4 + λ5)− q2(−1 + λ + 4λ2 − 3λ3 − 2λ4 + λ5)],


VU(N)[77] = (−1 + qλ)

(−1 + q)2q5(1 + q)λ3

[(−1 + λ)2λ + q12(−1 + λ)3λ2

− q(−1 + λ)2(1 + 2λ) − 2q11(−1 + λ)2λ2(−1 − λ + λ2)+ q3(−1 + λ)2(1 + 7λ + 2λ2)+ q5(−1 + λ)2(1 − 9λ − 8λ2 + 2λ3)+ q9(−1 + λ)2λ

(−1 − 8λ − λ2 + 2λ3)+ q2(2 − 5λ + λ2 + 4λ3 − 2λ4)− q7(−1 + λ)2(1 − 5λ − 11λ2 + 2λ4)+ q8λ

(−3 + 11λ − 16λ3 + 8λ4)+ q6(2 − 18λ2 + 15λ3 + 5λ4 − 4λ5)+ q4(−4 + 6λ + 8λ2 − 12λ3 + λ4 + λ5)+ q10λ

(1 − 2λ − 4λ2 + 8λ3 − λ4 − 3λ5 + λ6)],

VU(N)[81] = 1

(−1 + q)2q6(1 + q)λ7

[q(−1 + λ)2 − (−1 + λ)2λ − q4(−1 + λ)2λ

+ q3(−1 + λ)2λ2 + q5λ5(−1 + λ3)+ q11λ5(−1 + λ + λ3 − λ4)− q6λ4(−1 + λ − λ2 + λ4)+ q9λ5(−1 + λ − λ2 + λ4)+ q8λ4(−1 − λ + 2λ4)+ q10λ4(1 − λ + 2λ2 − λ3 − λ4 − λ5 + λ6)+ q7(λ4 + λ6 − λ8 − λ9)],

VU(N)[92] = 1

(−1 + q)2q9(1 + q)λ11

[q(−1 + λ)2 − (−1 + λ)2λ − q4(−1 + λ)2λ

+ q3(−1 + λ)2λ2 + q13λ6(−1 + λ4)+ q10λ6(−1 − λ + 2λ4)+ q9λ4(−1 + λ − λ2 + λ5)− q12λ5(−1 + λ − λ2 + λ5)+ q7λ5(1 − λ − λ4 + λ5)− q8λ4(1 − 2λ + λ2 − λ3 − λ4 + λ5 + λ6)+ q11(λ5 + λ7 − λ9 − λ10)],

VU(N)[102] = 1

(−1 + q)2q8(1 + q)λ9

[q(−1 + λ)2 − (−1 + λ)2λ − q4(−1 + λ)2λ

+ q3(−1 + λ)2λ2 + q7λ6(−1 + λ4)+ q13λ6(−1 + λ + λ4 − λ5)− q8λ5(−1 + λ − λ2 + λ5)+ q11λ6(−1 + λ − λ2 + λ5)+ q10λ5(−1 − λ + 2λ5)− q9λ5(−1 − λ2 + λ5 + λ6)+ q12λ5(1 − λ + 2λ2 − λ3 − λ5 − λ6 + λ7)].

There seems to be a symmetry transformation on the polynomial variables which gives the

U(N) invariants of knots carrying antisymmetric second rank tensor representation R = . Thesymmetry relation for these non-torus knots (see also Eq. (7) in [21])

VU(N)[K](q−1, λ

)= VU(N)[K](q,λ). (B.2)

We checked our polynomials for knots 62, 63, 72, 73 with the results obtained in Ref. [22]using character expansion approach. Before we use these polynomial invariants in verifying


Ooguri–Vafa conjecture, we shall enumerate the multicoloured link polynomials in the followingappendix.

Appendix C. Link polynomials

In this appendix we list the U(N) link invariant for non-torus links given in Fig. 2 for repre-

sentations R1,R2 ∈ { , , }.1. For R1 = , R2 = :

VU(N)( , ) [62] = 1

(−1 + q)2qλ

[λ(−1 + λ2)+ q4λ

(−1 + λ2)+ q(1 + λ − 2λ3)

+ q3(1 + λ − 2λ3)+ q2(−1 − 2λ + λ2 + 2λ3)],V

U(N)( , ) [63] = (−1 + λ)

(−1 + q)2qλ3

[λ + q4λ − q

(1 + 3λ + 2λ2)− q3(1 + 3λ + 2λ2)

+ q2(2 + 4λ + 3λ2 + λ3)],V

U(N)( , ) [71] = (−1 + λ)

(−1 + q)2q2λ2

[−λ − q6λ + q(1 + λ)2 + q5(1 + λ)2

− q2(2 + 3λ + λ2)+ q3(2 + 3λ + λ2)− q4(2 + 3λ + λ2)],V

U(N)( , ) [72] = (−1 + λ)

(−1 + q)2q2λ2

[−λ − q6λ + q(1 + λ)2 − 2q2(1 + λ)2 − 2q4(1 + λ)2

+ q5(1 + λ)2 + q3(2 + 5λ + 3λ2)],V

U(N)( , ) [73] = (−1 + λ)

(−1 + q)2qλ3

[−λ(1 + λ) − q4λ(1 + λ) + q(1 + λ)3 + q3(1 + λ)3

− q2(2 + 4λ + 5λ2 + λ3)].2. For R1 = , R2 = :

VU(N)( , )[62] = (−1 + λ)

(−1 + q)3√q(1 + q)λ3/2

[q − λ − q7λ3 + q8λ3 + q2λ

(1 + λ + λ2)

− q6λ(1 + λ + λ2)+ q5λ

(1 + 2λ + 2λ2)− q4(−1 + λ3)

− q3(1 + 2λ + λ2 + λ3)],V

U(N)( , )

[63] = (−1 + λ)

(−1 + q)3q5/2(1 + q)λ7/2

[−λ + q7λ2 − q2(1 + λ) + q5λ3(1 + λ)

+ q(1 + λ)2 − q6λ(1 + λ + 2λ2)+ q4(1 + 3λ + 2λ2 + λ3)

− q3(1 + 2λ + 2λ2 + 2λ3)],


VU(N)( , )[71] = 1

(−1 + q)3q5/2(1 + q)λ5/2

[q + q2(−1 + λ) + (−1 + λ)λ

− q10(−1 + λ)λ2 − qλ3 − q8(−1 + λ)λ3 + q7(1 + λ − 2λ3)− 2q4(−1 + λ3)+ q9λ

(−1 + λ3)+ q5(−1 + λ + λ3 − λ4)+ q3(−1 − λ + λ2 + λ4)+ q6(−1 − λ + λ3 + λ4)],

VU(N)( , )[72] = − (−1 + qλ)

(−1 + q)3q7/2(1 + q)λ5/2

[q + (−1 + λ)λ + q9(−1 + λ)λ − qλ2

+ q5(−1 + 3λ + λ2 − 3λ3)+ q8(λ − λ3)− q2(1 − 2λ + λ3)+ q4(2 + λ − 4λ2 + λ3)+ q7(1 − 2λ2 + λ3)+ q3(−1 − 3λ + 3λ2 + λ3)+ q6(−1 − 3λ + 3λ2 + λ3)],

VU(N)( , )[73] = 1

(−1 + q)3q3/2(1 + q)λ7/2

[q2(−1 + λ2)+ λ

(−1 + λ2)+ q4(1 + 2λ − 3λ4)+ q7(λ2 − λ4)− q

(−1 − λ + λ3 + λ4)+ q6λ

(−1 − λ + λ3 + λ4)+ q5(λ3 − λ5)+ q3(−1 − λ − 2λ2 + 2λ3 + λ4 + λ5)].

Interchanging R1, R2 on the two components of the link, gives the same polynomial:

VU(N)( , )[L] = V

U(N)( , )[L], (C.1)

and replacing the second rank symmetric representation by antisymmetric representation ,the link polynomials are related as follows:

VU(N)( , )[L](q−1, λ

)= −VU(N)

( , )[L](q,λ). (C.2)

3. For R1 = , R2 = :

VU(N)( , )[62] = 1

(−1 + q)4q(1 + q)2λ2

[−q2(−1 + λ)2 − (−1 + λ)2λ

+ q(−1 + λ)2(1 + λ) + q15λ4(−1 + λ2)− q13λ4(−2 + λ + λ2)+ q3(−1 + λ)2λ

(−2 + λ + λ2 + λ3)+ q12λ3(−2 − λ − 2λ2 + 5λ3)+ q11λ2(−1 + λ − 2λ2 + 4λ3 − 2λ4)+ q10λ

(−1 + 2λ + λ2 + 3λ3 − 5λ5)+ q8λ

(1 − 4λ + λ2 − 3λ3 + 4λ4 + λ5)

+ q5(−1 + 4λ − 4λ2 − λ4 + 2λ5)+ q9λ(1 + 2λ − 3λ2 − 5λ4 + 5λ5)

+ q7(1 − 4λ + λ2 + 2λ3 + 3λ4 + λ5 − 4λ6)+ q14(λ3 + λ5 − 2λ6)+ 2q4(1 − 2λ + λ3 + λ5 − λ6)+ q6(−1 + λ + 5λ2 − 4λ3 + λ4 − 5λ5 + 3λ6)],


VU(N)( , )[63] = (−1 + λ)(−1 + qλ)

(−1 + q)4q7(1 + q)2λ6

[(−1 + λ)λ + q12(−1 + λ)2λ2

+ q(1 + λ − 3λ2)+ q2(−2 + 4λ + λ2 − 2λ3)

+ q11λ2(−3 + 2λ + 3λ2 − 2λ3)+ q3(−1 − 5λ + 9λ2 + 3λ3)+ q8λ

(2 − 13λ − 5λ2 + 10λ3)+ q6(−2 + λ + 19λ2 − 2λ3 − 6λ4)

+ q4(4 − 5λ − 11λ2 + 5λ3 + λ4)+ q5(−1 + 9λ − 8λ2 − 12λ3 + 2λ4)+ q9λ

(2 + 7λ − 10λ2 − 3λ3 + 3λ4)

+ q10λ(−1 + 2λ + 6λ2 − 6λ3 − λ4 + λ5)

− q7(−1 + 7λ + 2λ2 − 17λ3 + λ4 + 2λ5)],V

U(N)( , )

[71] = (−1 + λ)(−1 + qλ)

(−1 + q)4q6(1 + q)2λ4

[(−1 + λ)λ − q18(−1 + λ)λ2

+ q(1 + λ − 2λ2)+ q17λ2(−2 + λ2)+ q3λ

(−4 + 3λ + λ2)+ q2(−2 + 2λ + λ2 − λ3)+ q7(5 + 5λ − 7λ2 + λ3)+ q4(3 + 2λ − 5λ2 + λ3)− q15λ

(−1 − 6λ + 2λ2 + λ3)+ q14λ

(3 − 5λ − 4λ2 + 2λ3)+ q13(1 − 5λ − 7λ2 + 5λ3)

+ q12(−2 − 3λ + 11λ2 + 2λ3 − 2λ4)+ q6(1 − 8λ + 3λ2 + λ3 − λ4)+ q11(−1 + 10λ + 2λ2 − 6λ3 + λ4)− q16(λ − 3λ3 + λ4)− q9(2 + 11λ − 5λ2 − 3λ3 + λ4)+ q5(−4 + 3λ + 2λ2 − 3λ3 + λ4)+ q8(−5 + 7λ + 6λ2 − 3λ3 + λ4)+ q10(5 − λ − 12λ2 + λ3 + λ4)],

VU(N)( , )[72] = (−1 + λ)(−1 + qλ)

(−1 + q)4q8(1 + q)2λ4

[(−1 + λ)λ − q18(−1 + λ)λ2

+ q(1 + λ − 2λ2)+ q17λ2(−2 + λ + λ2)+ q15(λ + 4λ2 − 6λ3)

+ q16λ(−1 + λ + 2λ2 − 2λ3)+ q2(−2 + 3λ + λ2 − λ3)

+ q3(−1 − 6λ + 5λ2 + λ3)+ q4(5 − 2λ − 9λ2 + 2λ3)+ q14λ

(2 − 8λ + 2λ2 + 5λ3)+ q11λ

(6 − 19λ − 4λ2 + 9λ3)

+ q7(5 − 13λ − 12λ2 + 12λ3)+ q9(−4 + 3λ + 25λ2 − 9λ3 − 5λ4)+ q13(1 − 5λ + 3λ2 + 10λ3 − 5λ4)+ q12(−2 + 3λ + 12λ2 − 14λ3 − 3λ4)+ q10(3 − 12λ + 20λ3 − 3λ4)+ q6(−5 − 5λ + 19λ2 + λ3 − 2λ4)+ q5(−2 + 13λ − 2λ2 − 6λ3 + λ4)+ q8(1 + 13λ − 17λ2 − 12λ3 + 5λ4)],


VU(N)( , )[73] = (−1 + λ)(−1 + qλ)

(−1 + q)4q6(1 + q)2λ6

[(−1 + λ)λ − q12λ3(4 − 4λ − 3λ2 + λ3)

+ q(1 + λ − 3λ2 + λ3)+ q13λ3(−1 − 3λ + λ2 + λ3)

+ q2(−2 + 4λ − 3λ3 + λ4)− q11λ2(2 − 8λ − 8λ2 + 5λ3 + λ4)− q3(1 + 5λ − 9λ2 + 3λ4)+ q9λ

(2 + 3λ − 20λ2 − 4λ3 + 7λ4)

+ q14(λ3 − λ5)+ q4(4 − 5λ − 8λ2 + 11λ3 + λ4 − λ5)− q8λ

(−2 + 13λ − 3λ2 − 22λ3 + λ4 + λ5)+ q5(−1 + 9λ − 10λ2 − 10λ3 + 9λ4 + λ5)+ q6(−2 + λ + 16λ2 − 13λ3 − 12λ4 + 2λ5)+ q10λ

(−1 + 4λ + 5λ2 − 15λ3 − 3λ4 + 2λ5)+ q7(1 − 7λ + 3λ2 + 22λ3 − 7λ4 − 5λ5 + λ6)].

Changing both the rank two symmetric representation by antisymmetric representation ,we find the following relation between the link polynomials:

VU(N)( , )

[L](q,λ) = VU(N)

( , )[L](q−1, λ

). (C.3)

With these polynomial invariants available for the non-torus knots and links in Figs. 1, 2, weare in a position to verify Ooguri–Vafa [7] and Labastida–Marino–Vafa [8] conjectures.

Appendix D. Reformulated link invariants

In this appendix we explicitly write the reformulated link invariant for the non-torus knots andlinks in Fig. 1 and Fig. 2. Rewriting the most general form of reformulated invariants fR[K] andfR1,R2[L] (see Eq. (2.21)) for representations and on the component knots, the expressionfor knots are

f [K] = V [K], (D.1)

f [K] = V [K] − 1

2

(V [K]2 + V

(2)[K]), (D.2)

f [K] = V [K] − 1

2

(V [K]2 − V

(2)[K]), (D.3)

where we have suppressed U(N) superscript on the knot invariants (VR[K] ≡ VU(N)R [K](q,λ)).

Further, we use the notation V(n)R [K] ≡ VR[K](qn, λn).

Ooguri–Vafa conjecture [7] states that the reformulated invariants for knots should have thefollowing structure

fR(q,λ) =∑s,Q

NQ,R,s

q1/2 − q−1/2aQqs, (D.4)

where NQ,R,s are integer and Q and s are, in general, half-integers. Clearly for R = (fun-

damental representation), the polynomial structure in Appendix B for VU(N)[K] (D.1) obeys

Eq. (D.4). We will verify for R = and in the following subsection.


The reformulated invariants in terms of two-component link invariants has the following form

for R1,R2 ∈ { , , }:f( , )[L] = V( , )[L] − V [K1]V [K2], (D.5)

f( , )[L] = V( , )[L] − V( , )[L]V [K2] − V [K1]V [K2] + V [K1]V [K2]2, (D.6)

f( , )

[L] = V( , )

[L] − V( , )[L]V [K2] − V [K1]V [K2] + V [K1]V [K2]2, (D.7)

f( , )[L] = V( , )[L] − V( , )[L]V [K1] − V [K1]V [K2] + V [K1]2V [K2], (D.8)

f( , )

[L] = V( , )

[L] − V( , )[L]V [K1] − V [K1]V [K2] + V [K1]2V [K2], (D.9)

f( , )[L] = V( , )[L] − V (K1)V [K2] − V( , )[L]V [K2]− V( , )[L]V [K1] − 1

2V( , )[L]2 + 2V( , )[L]V [K1]V [K2]

+ V [K1]2V [K2] + V [K1]V [K2]2 − 3

2V( )[K1]2V( )[K2]2

− 1

2V

(2)( , )[L] + 1

2V

(2)[K1]V (2)[K2], (D.10)

f( , )

[L] = V( , )

[L] − V [K1]V [K2] − V( , )

[L]V [K2]− V

( , )[L]V [K1] − 1

2V( , )[L]2 + 2V( , )[L]V [K1]V [K2]

+ V [K1]2V [K2] + V [K1]V [K2]2 − 3

2V( )[K1]2V( )[K2]2

− 1

2V

(2)( , )[L] + 1

2V

(2)[K1]V (2)[K2]. (D.11)

Here the components knots K1 and K2 are unknots for the non-torus links in Fig. 2. The gener-alisation of Ooguri–Vafa conjecture for links was proposed in [8] which states that reformulatedinvariants for r-component link should have the following structure

f(R1,R2,...,Rr )(q, λ) = (q1/2 − q−1/2)r−2∑Q,s

N(R1,...,Rr ),Q,sqsλQ, (D.12)

where N(R1,...,Rr ),Q,s are integer and Q and s are half-integers.We can see below that all the reformulated invariants we calculate indeed satisfy the conjec-

ture.

D.1. Reformulated invariant for knots

We have already seen in Appendix B, V [K] has the Ooguri–Vafa form given in Eq. (D.4).For the symmetric second rank tensor R = placed on the knot, f [K] are:

f [41] = (−1 + λ)2

(−1 + q)q2λ3

[−q + λ − q5λ3 + q4λ4 + q3λ(1 + λ) − q2λ2(1 + λ)],

f [52] = − (1 − q + q2)(−1 + λ)2

(−1 + q)q5λ7

[q(−1 + λ) + q2(−1 + λ) + λ + q4λ

(1 + λ + λ2)

− q3(1 + λ2 + λ3 + λ4)],


f [61] = − (−1 + λ)2

(−1 + q)q4λ5

[q − λ − q2λ + q7λ4(1 + λ) + q3(1 + λ2)

− q5λ(1 + λ + λ2)2 + q4λ

(−1 + λ + 2λ2 + 2λ3 + λ4)+ q6(λ2 + λ3 + λ4 − λ5 − λ6)],

f [62] = − (1 − q + q2)(−1 + λ)2

(−1 + q)q6λ5

[−λ − q5λ + q2(−1 + λ)λ + q9λ3 − q8(−1 + λ)λ3

+ q6λ2(1 + λ) + q(1 + λ2)− q3λ

(2 + λ2)+ q4(1 + 2λ2)

− q7λ(1 + λ + λ3)],

f [63] = − (1 − q + q2)(−1 + λ)2

(−1 + q)q5λ3

[−λ + q2λ2 − q7λ2 + q9λ3 + q(1 + λ2)

+ q3(−1 − λ + λ2)+ q6λ2(−1 + λ + λ2)− q4λ(2 + 2λ + 2λ2 + λ3)

+ q5(1 + 2λ + 2λ2 + 2λ3)− q8(λ2 + λ4)],f [72] = − (−1 + λ)2

(−1 + q)q7λ9

[q − λ − q2λ − 2q4λ + q3(1 + λ2)

− q8λ(1 + λ + 2λ2 + 2λ3 + λ4)− q6λ

(2 + λ + 3λ2 + 3λ3 + 3λ4 + λ5)

+ q5(1 + λ2 + λ4 + λ5 + λ6)+ q7(1 + λ + 3λ2 + 3λ3 + 4λ4 + 3λ5 + λ6)],f [73] = (−1 + λ)2λ3

(−1 + q)q2

[−2q10λ3 − q12λ3 + q11λ4 − λ(1 + λ + λ2)

+ q6λ(−1 − λ − 5λ2 + λ3)+ q4λ

(−1 − λ − 4λ2 + λ3)− 2q2(1 + 2λ + 2λ2 + 2λ3)+ q5λ

(1 + 2λ − λ2 + 3λ3)

+ q8(−1 − λ − λ2 − 3λ3 + λ4)+ q(1 + 3λ + 3λ2 + λ3 + λ4)

+ q3(2 + 3λ + 4λ2 + 2λ4)+ q9(1 + λ + λ2 − λ3 + 2λ4)+ q7(1 + 2λ + 2λ2 − λ3 + 2λ4)],

f [74] = (−1 + λ)2λ

(−1 + q)q

[−q9λ5 + λ2(1 + λ) − q7λ3(1 + λ) + q8λ4(−1 + λ2)+ q6λ2(1 + λ + λ2 + λ3 + λ4)− qλ

(2 + 4λ + 4λ2 + 3λ3 + λ4)

+ q5(λ + λ2 − 2λ4 − λ5)− q3(2 + 5λ + 5λ2 + 5λ3 + 4λ4 + λ5)+ q4(1 + λ + 2λ2 + λ3 − λ4 + λ5 + λ6)+ q2(1 + 5λ + 8λ2 + 7λ3 + 3λ4 + 2λ5 + λ6)],

f [75] = (1 − q + q2)(−1 + λ)2

(−1 + q)q9λ9

[−q + λ − q7(−1 + λ)λ3 + q8λ(1 + λ)3

+ q10λ(1 + 2λ + 2λ2)+ q4(−1 + λ + λ2 + 2λ3)+ q6(λ3 − 2λ4)

+ q3(λ − λ4)− q5(1 + λ + λ2 + λ4)− q9(1 + 2λ + 3λ2 + 3λ3 + 2λ4)],


f [76] = − (−1 + λ)2

(−1 + q)q6λ7

[−(−1 + λ)λ2 + q11λ5 − q10λ5(1 + λ)

+q9λ3(−2 − 2λ − λ2 + λ3)+ qλ(−2 + λ + λ2 + λ3)

− q3λ(1 − 3λ + λ2 + λ4)− q2(−1 + λ2 + 3λ3 + λ4)

+ q8λ2(2 + 4λ + 5λ2 + 4λ3 + λ4)− q5λ(3 + 4λ + 8λ2 + 6λ3 + 2λ4)

− q7λ(2 + 2λ + 7λ2 + 6λ3 + 4λ4 + 2λ5)

+ q4(1 − λ + 4λ2 + 3λ3 + 5λ4 + 2λ5)+ q6(1 + λ + 6λ2 + 6λ3 + 7λ4 + 2λ5 + λ6)],

f [77] = (−1 + λ)2

(−1 + q)q5λ3

[λ + 2q3λ + q11(−1 + λ)λ3 − q(1 + λ)2 + q2(2 + λ)

− q8λ2(−1 + λ − 2λ2 + λ3)+ q9λ2(−1 − 2λ − λ2 − λ3 + λ4)− q5λ

(7 + 12λ + 12λ2 + 5λ3 + 2λ4)+ q4(−2 + 5λ2 + 5λ3 + 4λ4)

+ q10(λ2 + λ3 + 2λ4 − 2λ5)+ q6(2 + 8λ + 11λ2 + 9λ3 + 4λ4 − λ5)+ q7(−1 − 3λ − 4λ2 − 3λ3 + 3λ4 − λ5 + λ6)],

f [81] = − (−1 + λ)2

(−1 + q)q6λ7

[q − λ − q2λ − 2q4λ + q3(1 + λ2)+ q5(1 + λ2)

+ q9λ5(1 + λ + λ2)− q8λ2(−1 − 2λ − 2λ2 − 2λ3 + λ5 + λ6)+ q6λ

(−1 + λ + 2λ2 + 3λ3 + 3λ4 + 2λ5 + λ6)− q7λ

(1 + 2λ + 4λ2 + 5λ3 + 4λ4 + 3λ5 + λ6)],

f [92] = − (−1 + λ)2

(−1 + q)q9λ11

[q − λ − q2λ − 2q4λ − 2q6λ + q3(1 + λ2)+ q5(1 + λ2)

− q10λ(1 + λ + 2λ2 + 3λ3 + 3λ4 + 2λ5 + λ6)

− q8λ(2 + λ + 2λ2 + 4λ3 + 4λ4 + 4λ5 + 3λ6 + λ7)

+ q7(1 + λ2 + λ5 + λ6 + λ7 + λ8)+ q9(1 + λ + 3λ2 + 4λ3 + 5λ4 + 6λ5 + 5λ6 + 3λ7 + λ8)],

f [101] = − (−1 + λ)2

(−1 + q)q8λ9

[q − λ − q2λ − 2q4λ − 2q6λ + q3(1 + λ2)+ q5(1 + λ2)

+ q7(1 + λ2)+ q11λ6(1 + λ + λ2 + λ3)+ q10λ2(1 + 2λ + 3λ2 + 3λ3 + 3λ4 + λ5 − λ7 − λ8)+ q8λ

(−1 + λ + 2λ2 + 3λ3 + 4λ4 + 4λ5 + 3λ6 + 2λ7 + λ8)− q9λ

(1 + 2λ + 4λ2 + 6λ3 + 7λ4 + 6λ5 + 5λ6 + 3λ7 + λ8)].

Changing the symmetric representation by antisymmetry representation, we find the followingrelation between the reformulated invariants:

f [K](q−1, λ)= f [K](q,λ). (D.13)


D.2. Reformulated invariant for links

1. For R1 = , R2 = :

f( , )[62] = (−1 + λ)

qλ

[−q + λ + q2λ + λ2 + q2λ2],f( , )[63] = (−1 + λ)

qλ3

[λ + q2λ − q

(1 + λ + 2λ2)],

f( , )[71] = (−1 + λ)

q2λ2

[−λ − q4λ + q2(−2 + λ)λ + q(1 + λ2)+ q3(1 + λ2)],

f( , )[72] = (−1 + λ)

q2λ2

[−λ − 3q2λ − q4λ + q(1 + λ2)+ q3(1 + λ2)],

f( , )[73] = (−1 + λ2)

qλ3

[q − λ − q2λ + qλ2].

2. For R1 = , R2 = :

f( , )[62] = (−1 + λ)√q√

λ

[λ2 + qλ2 + q3λ2 + q4λ2 + q2(−1 − λ + λ2)],

f( , )[63] = 1

q5/2λ7/2(1 + q)(−1 + λ)

[λ + q2λ − q

(1 + λ + λ2)],

f( , )[71] = (−1 + λ)

q5/2λ5/2

[q − λ − q6λ2 + q3(−2 + λ)λ2 + q2λ3

+ q4λ(1 − λ + λ2)+ q5λ

(1 − λ + λ2)],

f( , )[72] = (−1 + λ)

q7/2λ5/2

[q4 − λ − 3q3λ − q6λ2 + q5λ3 + q

(1 − λ + λ2)

+ q2(1 − λ + λ2)],f( , )[73] = (−1 + λ2)

q3/2λ7/2

[q − λ − q3λ2 + q2λ3].

We have checked that f( , )[L] = f( , )[L] for these links. We also have the symmetryrelation

f( , )[L](q−1, λ)= −f

( , )[L](q,λ). (D.14)

3. For R1 = , R2 = :

f( , )[62] = 1

q2λ

[q(−1 + λ)2λ + q9(−1 + λ)2λ2 + λ3 − λ5

+ q7(−1 + λ)2λ2(3 + λ) + q10λ3(−1 + λ2)+ q8λ2(2 − 3λ + λ2)+ q5(−1 + λ)2(−1 − 2λ + 2λ2)+ q3(−1 + λ)2(1 + λ + 2λ2)+ q4(−1 + λ + 4λ2 − 5λ3 + λ4)+ q6λ

(−1 + 5λ − 8λ2 + 3λ3 + λ4)+ q2(−1 + λ2 + λ4 − λ5)],


f( , )[63] = 1

q7λ6

[−(−1 + λ)2λ + q2(−1 + λ)2λ2 − q9(−1 + λ)λ3

+ q5(−1 + λ)2λ2(2 + λ2)− q4(−1 + λ)2λ(1 − λ + 2λ2)

+ q8(−1 + λ)2λ(1 + λ + 2λ2)+ q3λ2(3 − 4λ + 3λ2 − 2λ3)

+ q6(−1 + λ)2λ(1 + 3λ + 2λ2 + 2λ3)+ q

(1 − 2λ + λ2 − λ3 + λ4)

− q7(1 + 2λ2 − 2λ3 − 2λ4 + λ6)],f( , )[71] = 1

q6λ4

[−(−1 + λ)2λ + q11(−1 + λ)4λ2 − q13(−1 + λ)2λ3

+ q(−1 + λ)2(1 + λ2)+ 3q9(−1 + λ)2λ2(1 − λ + λ2)+ q3(−1 + λ)2(1 − λ + 2λ2)+ q2λ

(−2 + 4λ − 3λ2 + λ3)+ q12λ2(1 − 2λ + 3λ2 − 3λ3 + λ4)− q6(1 + λ − λ2 − 2λ3 + λ4)+ q7(−1 + λ)2(1 + 2λ + 2λ2 − λ3 + λ4)+ q8λ2(2 − 4λ + 7λ2 − 7λ3 + 2λ4)− q5λ

(−1 + λ + 2λ3 − 3λ4 + λ5)+ q10λ

(−1 + 2λ − 7λ2 + 11λ3 − 7λ4 + 2λ5)− q4(−1 + 3λ − 4λ2 + λ3 + λ4 − λ5 + λ6)],

f( , )[72] = − 1

q8λ4

[(−1 + λ)2λ + q13(−1 + λ)2λ3

− q(−1 + λ)2(1 + λ2)+ q3(−1 + λ)2λ(3 − 2λ + λ2)

+ q11(−1 + λ)2λ(−1 − λ + λ2)+ q5(−1 + λ)2(−1 + 2λ − 5λ2 + λ3)

− q2λ(−1 + 4λ − 4λ2 + λ3)− q12λ3(−1 + 2λ − 2λ2 + λ3)

− q7(−1 + λ)2(1 + λ + 5λ2 + λ3)+ q10λ2(1 − 2λ − 2λ2 + 3λ3)+ q6λ

(3 − 10λ + 9λ2 − 5λ3 + 3λ4)

+ q4(−1 + 6λ − 11λ2 + 12λ3 − 7λ4 + λ5)− q9λ

(3 − 3λ − 2λ3 + λ4 + λ5)

+ q8(1 + λ + 2λ2 − 5λ3 + λ4 − λ5 + λ6)],f( , )[73] = − (−1 + λ)2

q6λ6

[−q + λ − q5λ + q9λ4(1 + λ) + q4λ(1 + λ)2

+ q6λ2(1 + λ)2(1 + λ2)− q3λ2(1 + λ + λ2)− q7λ

(1 + 2λ + 3λ2 + 3λ3 + 3λ4 + λ5)+ q8(λ2 + λ3 + 2λ4 − λ6)].

Changing both the rank two symmetric representation by antisymmetric representation ,we find the following relation between the reformulated invariants:

f( , )[L](q,λ) = f( , )

[L](q−1, λ). (D.15)


References

[1] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351, http://dx.doi.org/10.1007/BF01217730.

[2] P. Rama Devi, T. Govindarajan, R. Kaul, Three-dimensional Chern–Simons theory as a theory of knots andlinks. 3. Compact semisimple group, Nucl. Phys. B 402 (1993) 548–566, arXiv:hep-th/9212110, http://dx.doi.org/10.1016/0550-3213(93)90652-6.

[3] P. Ramadevi, Chern–Simons theory as a theory of knots and links, PhD thesis, The Institute of Mathematical Sci-ences (IMSc), 1996.

[4] J. Labastida, P. Llatas, A. Ramallo, Knot operators in Chern–Simons gauge theory, Nucl. Phys. B 348 (1991) 651–692, http://dx.doi.org/10.1016/0550-3213(91)90209-G.

[5] A.N. Kirillov, N.Y. Reshetikhin, Representation algebra Uq(sl2), q-orthogonal polynomials and invariants of links,in: T. Kohno (Ed.), New Developments in the Theory of Knots, World Scientific, Singapore, 1989.

[6] R. Kaul, Chern–Simons theory, colored oriented braids and link invariants, Commun. Math. Phys. 162 (1994) 289–320, arXiv:hep-th/9305032, http://dx.doi.org/10.1007/BF02102019.

[7] H. Ooguri, C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419–438, arXiv:hep-th/9912123, http://dx.doi.org/10.1016/S0550-3213(00)00118-8.

[8] J. Labastida, M. Marino, C. Vafa, Knots, links and branes at large N , JHEP 0011 (2000) 007, arXiv:hep-th/0010102.[9] P. Ramadevi, T. Sarkar, On link invariants and topological string amplitudes, Nucl. Phys. B 600 (2001) 487–511,

arXiv:hep-th/0009188, http://dx.doi.org/10.1016/S0550-3213(00)00761-6.[10] E. Witten, Fivebranes and knots, arXiv:1101.3216.[11] A. Brini, B. Eynard, M. Marino, Torus knots and mirror symmetry, arXiv:1105.2012.[12] M. Aganagic, S. Shakirov, Knot homology from refined Chern–Simons theory, arXiv:1105.5117.[13] D. Gaiotto, E. Witten, Knot invariants from four-dimensional gauge theory, arXiv:1106.4789.[14] M. Marino, C. Vafa, Framed knots at large N , arXiv:hep-th/0108064.[15] P. Borhade, P. Ramadevi, T. Sarkar, U(N) framed links, three manifold invariants, and topological strings, Nucl.

Phys. B 678 (2004) 656–681, arXiv:hep-th/0306283, http://dx.doi.org/10.1016/j.nuclphysb.2003.11.023.[16] J.S. Birman, Braids, Links and Mapping Class Groups, Annals of Mathematics Studies, Princeton Univ. Press,

Princeton, NJ, 1975.[17] J.M.F. Labastida, M. Marino, A New point of view in the theory of knot and link invariants, arXiv:math/0104180.[18] C. Paul, P. Borhade, P. Ramadevi, Composite representation invariants and unoriented topological string amplitudes,

Nucl. Phys. B 841 (2010) 448–462, arXiv:1008.3453, http://dx.doi.org/10.1016/j.nuclphysb.2010.08.013.[19] P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, et al., A new polynomial invariant of knots and links,

Bull. Am. Math. Soc. 12 (1985) 239–246, http://dx.doi.org/10.1090/S0273-0979-1985-15361-3.[20] J.H. Przytycki, K.P. Traczyk, Conway algebras and skein equivalence of links, Proc. Amer. Math. Soc. 100 (1987)

744–748.[21] H. Itoyama, A. Mironov, A. Morozov, An. Morozov, HOMFLY and superpolynomials for figure eight knot in all

symmetric and antisymmetric representations, JHEP 1207 (2012) 131, arXiv:1203.5978, http://dx.doi.org/10.1007/JHEP07(2012)131.

[22] H. Itoyama, A. Mironov, A. Morozov, An. Morozov, Character expansion for HOMFLY polynomials. III. All3-Strand braids in the first symmetric representation, Int. J. Mod. Phys. A 27 (2012) 1250099, arXiv:1204.4785,http://dx.doi.org/10.1142/S0217751X12500996.

http://dx.doi.org/10.1007/BF01217730

http://dx.doi.org/10.1016/0550-3213(93)90652-6

http://dx.doi.org/10.1016/0550-3213(93)90652-6

http://dx.doi.org/10.1016/0550-3213(91)90209-G

http://dx.doi.org/10.1007/BF02102019

http://dx.doi.org/10.1016/S0550-3213(00)00118-8

http://dx.doi.org/10.1016/S0550-3213(00)00761-6



http://dx.doi.org/10.1090/S0273-0979-1985-15361-3

http://dx.doi.org/10.1007/JHEP07(2012)131


http://dx.doi.org/10.1142/S0217751X12500996

http://dx.doi.org/10.1007/BF01217730

JHEP11(2011)111

Published for SISSA by Springer

Received: September 6, 2011

Accepted: October 29, 2011

Published: November 23, 2011

Inverse algorithm and M2-brane theories

Siddharth Dwivedia and Pichai Ramadevia,b

aDepartment of Physics, Indian Institute of Technology Bombay,Mumbai, IndiabCenter for Quantum Spacetime, Sogang University,Seoul, South Korea

E-mail: [email protected], [email protected]

Abstract: Recent paper arXiv:1103.0553 studied the quiver gauge theories on coincidentM2 branes on a singular toric Calabi-Yau 4-folds which are complex cone over toric Fano3-folds. There are 18 toric Fano manifolds but only 14 toric Fano were obtained fromthe forward algorithm. We attempt to systematize the inverse algorithm which helps inobtaining quiver gauge theories on M2-branes from the toric data of the Calabi-Yau 4-folds.In particular, we obtain quiver gauge theories on coincident M2-branes corresponding tothe remaining 4 toric Fano 3-folds. We observe that these quiver gauge theories cannot begiven a dimer tiling presentation.

Keywords: AdS-CFT Correspondence, M-Theory

ArXiv ePrint: 1108.2387

c© SISSA 2011 doi:10.1007/JHEP11(2011)111



http://arxiv.org/abs/1108.2387


JHEP11(2011)111

Contents

1 Introduction 1

2 Toric data G → quiver Chern-Simons 22.1 Forward algorithm 32.2 Inverse algorithm 4

3 M2-brane theories from CY 4-folds with b2 = 1 53.1 C4/Z2 63.2 Fano P3 theory 7

4 M2-brane theories from CY 4-folds with b2 = 2 84.1 Fano B4 theory 94.2 Fano B1 theory 11

4.2.1 Hilbert series for B1 theory 134.3 Fano B2 theory 15

4.3.1 Hilbert series evaluation for B2 theory 184.3.2 Genus for B2 theory 18

4.4 Fano B3 theory 194.4.1 Hilbert series evaluation for B3 theory 21

5 Summary and open problems 22

1 Introduction

Starting with the works of Bagger-Lambert [1–3], Gustavsson [4, 5], Raamsdonk [6] andAharony-Bergman-Jafferis-Maldacena (ABJM) [7], we see interesting developments in thelast three years between supersymmetric Chern-Simons gauge theory on coincident M2branes at the tip of Calabi-Yau 4-folds and their string duals. For a nice review, see ref. [8].

For a class of the supersymmetric Chern-Simons theories which can be represented bya quiver diagram, AdS4/CFT3 correspondence has been studied [9]. The Calabi-Yau 4-fold toric data can be obtained for the quiver Chern-Simons theories by a procedure calledforward algorithm. This approach was initially studied to obtain toric data of Calabi-Yau 3-folds from 3+1-dimensional quiver supersymmetric theories [10]. Further, if thequiver data can admit dimer tilings [11, 12], then the toric data can be obtained fromthe determinant of Kastelyne matrix. Generalising the forward algorithm/dimer tilingprocedure to 2+1-dimensional quiver Chern-Simons theories resulted in obtaining toricdata of many Calabi-Yau 4-folds [13–20].

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In the recent paper [20], toric data, genus, second Betti number of 18 toric Fano3-folds have been tabulated. We believe that there must be at least one quiver Chern-Simons theory corresponding to every toric Calabi-Yau 4-fold which are complex conesover these Fano 3-folds. Using the forward algorithm and tilings [20], the toric data of 14Fano 3-folds were obtained from the corresponding quiver Chern-Simons theories. Findinga quiver Chern-Simons corresponding to the remaining four toric Fanos P3,B1,B2,B3 is achallenging problem which we try to attempt by systematizing the inverse algorithm.

Reversing the procedure of the forward algorithm, called inverse algorithm, shouldresult in obtaining quiver gauge theories from the toric data. As already pointed out inthe context of Calabi-Yau 3-folds [10], the inverse algorithm has ambiguities which we willdetail in section 2. Considering the toric Calabi-Yau 3-folds as embeddings inside the non-cyclic orbifolds C3/(Zn × Zm) and performing partial resolutions, the matter content andthe superpotential W of the 3+1 quiver theories were obtained [10]. However, the adjointmatter fields could not be explained by this method. The adjoint fields appear naturally inthe algebraic approach [21] which involves matching matrix corresponding to dimer tiling.

Exhaustive works [13]-[20] show that all the studied quiver Chern-Simons theoriescorresponding to the Calabi-Yau 4-folds admit dimer tiling. Further, using higgsing [19]of matter fields on a known G-node quiver which admits tiling, (G − 1) node quiver andtheir corresponding toric data were obtained. In fact, the approach [21] can be extendedto 2 + 1-dimensional quiver gauge theories giving the results in ref. [19]. All the quivertheories before or after higgsing can be represented as tiling. Unfortunately P3,B1,B2,B3

toric data have not been obtained from the higgsing approach. So, we believe that thesefour Fano 3-folds may not give quivers admiting tiling description. It is also not clearwhether we can perform partial resolution of an abelian orbifold of C4 [10] and obtain toricdata of these Fano 3-folds.

One of the crucial step in the inverse algorithm is to fix the F-term and D-term chargeassignments corresponding to the toric data. In this work, we try to understand the patternof the F-term and the D-term charges for the 14 Fano 3-folds whose quivers are known.With this pattern identification, we propose (see ansatz in section 4.1) a form for thesecharges. Then, the rest of the sequence of the inverse algorithm can be performed to givethe quiver data.

The plan of the paper is as follows: In section 2, we briefly review the forward andthe inverse algorithm. In section 3, we first review the inverse algorithm of C4/Z2 whichclosely resembles Fano P3 and then derive the quiver and the mesonic moduli space Hilbertseries for Fano P3. In section 4, we first review inverse algorithm for Fano B4 whose quiveris known from forward algorithm. This helps in understanding the charge assignments forthe other three Fanos B1,B2,B3. We then present the details of quiver and Hilbert seriesfor these three Fanos in later subsections. Finally, we summarize in section 5.

2 Toric data G → quiver Chern-Simons

We will briefly discuss the inverse algorithm which is used to obtain quiver gauge theoriesfrom the toric data G4×c describing the Calabi-Yau (CY) 4-folds. Here c denotes the

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number of points (including multiplity of points) in the toric diagram. Unlike the forwardalgorthim, which starts from the quiver data and superpotential W giving a unique toricdata, the inverse algorithm is non-unique. That is, there can be many quiver gauge theoriespossible from the inverse algorithm. Besides this non-uniqueness, there are many subtleambiguities in choosing the toric data. They are:

1. Two toric data G and G′ are equivalent if they are related by any GL(4,Z) transfor-mation T , that is, G = T .G′. This leads to a huge pool of possible nullspace of G -namely, the charge matrix Q(c−4)×c satisfying Q.Gt = 0 can be many.

2. The multiplicity of the toric points gives the toric data with repeated columns butthey represent the same Calabi-Yau 4-folds. Usually, it is not clear which pointswith what multiplicity in the toric diagram to be taken. We could start with nomultiplicity of all the toric points and if we end with exotic or insensible quivers, wecould try putting multiplicity of some toric points. This is definitely very tedious.

We try to resolve some of these ambiquities by understanding the pattern of the matrixQ(c−4)×c of the 14 toric Fano 3-folds derived from the forward algorithm [20]. We can obtainthe steps of the inverse algorithm by reversing the sequence of steps in the forward algorthm.

2.1 Forward algorithm

We will now recapitulate the essential aspects of the forward algorithm where one startswith a N = 2 Chern-Simons (CS) quiver gauge theory and superpotential W . For toricquivers, there are NT terms in W with each matter field appearing only in two terms withopposite signs.

1. From the quiver data represented as a quiver diagram, we know the number ofgauge groups G (number of nodes), CS levels ka for each node and m number ofbi-fundamental matter fields and adjoints fields Xi’s. From this diagram, we canwrite the quiver charge matrix elements dai where the index a = 1, 2, . . . G andi = 1, 2, . . .m. In the CS quivers, besides

∑a dai = 0, we also require that the CS

levels ka’s have GCD({ka}) = 1 and∑G

a=1 ka = 0 (Calabi-Yau requirement). Usingthe above conditions and the equation (moment map) of U(1)G abelian CS quivergauge theories,

µa(X) =∑i

dai|Xi|2 = kaσ , (2.1)

where σ is the scalar component of the vector superfield Va, we can obtain only(G−2) D-term equations giving a projected charge matrix ∆(G−2)×m from the matrixelements dai. That is, the matrix elements of projected charge matrix satisfies∑

i

∆bi|Xi|2 = 0 , (2.2)

where b = 1, 2, . . . (G − 2). For example, take a G = 3 node quiver with the the CSlevels k1, k2,−(k1 +k2) where GCD(k1, k2) = 1. Further the eqs. (2.1), (2.2) suggeststhat the projected charge is a single row matrix whose elements are given by

∆i = k2d1i − k1d2i . (2.3)

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2. From the F-term constraint equation ∂W/∂Xi = 0, we can obtain relations betweenmatter fields. Introducing (G + 2) fields vr’s, we can incorporate the F -term con-straints in the matrix Kir which relates the matter fields to vi in the following way:

Xi =∏r

vKirr (2.4)

The dual of the K-matrix satisfying K.T ≥ 0 (all entries of the matrix K.T arenon-negative) will give a matrix T(G+2)×c where c gives the number of GLSM sigmamodel fields pα’s.

3. From the K and T , we can write a matrix P = K.T . The entries of all these matricesK, T , P are integers. For quiver theories which can admit tiling, one can read offP -matrix from W . The P -matrix relates the matter fields to GLSM pα fields as

Xi =∏α

pPiαα . (2.5)

The kernel of the T as well as P (T.QtF = P.QtF = 0) will give the QF (c−G−2)×ccharge matrix.

4. From the relation (2.5), we can obtain the baryonic charge matrixQD(G−2)×c elementsfrom the projected charge (2.2) matrix elements as follows:

∆bi =∑α

Piα(QD)bα . (2.6)

5. The total Q

Q(c−4)×c =

(QF (c−G−2)×cQD(G−2)×c

), (2.7)

whose kernel Q.Gt = 0 gives the toric data G4×c.

2.2 Inverse algorithm

Now, we can reverse the sequence and try to obtain the quiver CS theory on M2 branesat the tip of singular CY 4-folds described by G. There are additional data of toric Fanowhich are useful to handle some of the ambiguities we had enumerated.

1. For the toric Fano 3-folds, G can be written in a form where the symmetry of thecorresponding CY 4-fold SU(4)i1 × SU(3)i2 × SU(2)i3 × U(1)i4 is seen as the simpleroots along the rows of G. As the rank of CY is 4, we require 3i1 + 2i2 + i3 + i4 = 4.

2. The second betti number b2 of the toric Fano 3-fold is related to the number ofexternal points (E) in the toric diagram as: b2 = E − 3. Further the number ofbaryonic symmetries for the Fano is: b2 − 1 = E − 4. This fixes that the number ofrows in the QD matrix must be

b2 − 1 = G− 2 . (2.8)

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JHEP11(2011)111

3. With the GL(4,Z) freedom, it is not obvious as to what G′ we have to choose andfind the nullspace Q satisfying Q.G′t = 0. However, from the eq. (2.8), we know howmany rows must represent QD matrix. Equivalently, we must find a CS quiver withG = b2 + 1.

4. We try to understand the pattern of Q for a toric Fano with same b2 obtained fromforward algorithm and implement the same for the missing Fano 3-folds. Besides b2,the symmetry of the Fano also suggests how to incorporate the charge assignments inthe Q matrix. Further, the multiplicity of points in the toric diagram is also suggestedby this pattern identification. Following this methodology, we could guess a form forQF . Using QF , we determine T and hence K.

5. The number of rows of the matrix K gives the number of matter fields. From K, wefind a relation between matter fields which are supposed to be F -term constraints.For the G = (b2 +1) node quiver with the F-term constraints on the matter fields, wetry to reconstruct all possible toric quiver superpotential W . Then we can draw thequiver diagram where the terms in W must denote closed cycles in the quiver diagram.

6. Using the pattern for QD from the forward algorithm for the known Fano, we caninfer QD charge assignment pattern for other Fano 3-folds. Further, the QD mustsatisfy projected quiver charge (2.6). This is the non-trivial part but for small G,it is not difficult to find the choice of the CS levels which will give (2.3) satisfyingeq. (2.6).

In the following two sections, we will first work out the inverse algorithm for two Calabi-Yau 4-folds whose quivers are known from the tiling/forward algorithm. This will help toundersand the pattern in choosing the charge matrix QF and QD for the unknown FanosP3,B1,B2,B3. Using this pattern, we then perform the inverse algorithm for the missingFano 3-folds and obtain the corresponding toric quiver CS theories. We also compute theHilbert series and the R-charge assignments.

3 M2-brane theories from CY 4-folds with b2 = 1

From the dimer tilings, we know that the number of gauge-groups is G = 2 for the quivergauge theory corresponding to orbifolds of C4 (C4/Zk) which from eq. (2.8) implies b2 = 1.The toric Fano 3-fold P3 also has b2 = 1. Both C4/Zk and P3 have same number of externalpoints in the toric diagram and hence zero baryonic symmetry: b2 − 1 = 0. That is, QDis zero. So, we expect to obtain a G = 2 node quiver for the toric Fano P3 using inversealgorithm. For these Calabi-Yau 4-folds whose QD = 0, we can directly take the matrixT = G because QF .T t = QF .Gt = 0. Adding multiplicity of points in the toric diagram willresult in repetition of some columns of the T -matrix which will not alter the K-matrix.So, adding multiplicity of points in the toric diagram will not change the quiver data forb2 = 1 CY 4-folds. We will first implement inverse algorithm for the orbifold C4/Z2 as awarm-up exercise and then obtain 2-node quiver for P3 Fano 3-fold.

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JHEP11(2011)111

3.1 C4/Z2

Let us the take the toric data G for C4/Z2 obtained from the tiling approach (see eq. (3.2)in ref. [18]):

G =

1 1 1 1−1 0 −1 00 −1 −1 00 0 2 0

(3.1)

As QD = 0, we can take T = G. The matrix K from T will be0 0 0 10 0 −2 −10 −2 0 −12 2 2 1

(3.2)

The row index of the 4×4 square matrix K indicates that the number of matter fields in thequiver theory is 4 confirming the known data for C4 orbifolds. Also, being a square matrix,it is not possible to find F -term constraint equations. From the matter field content, onecan try to construct all possible connected quiver diagrams and the loops in the quiverdiagram will give gauge invariant terms in superpotential W . The information about theChern-Simons level ka is inferred from the matrix P = K.T which for this case is given as:

P =

0 0 2 00 2 0 02 0 0 00 0 0 2

(3.3)

For C4, the P matrix from inverse algorithm will turn out to have entries zero or 1. So,the entries for C4/Z2 which are 2 or 0 indicates that one of the nodes of the 2-node quiverdiagram must have level k1 = 2 and the other node by Calabi-Yau requirement has levelk2 = −k1 = −2. Inferring the CS levels from P -matrix is only applicable for CY 4-foldswhose QD = 0. The QF satisfying P.QtF = 0 is also trivial.

Hence from the inverse algorithm for C4/Z2, there can be three possible 2-node quiverswith four matter fields and levels k1 = −k2 = 2. They are:

1. ABJM theory with 4 bi-fundamental matter fields Xi12 and Xi

21 where i = 1, 2 andW = Tr[εijX1

12Xi21X

212X

j21] whose abelian W = 0.

2. 2-node quiver with two adjoints φ12, φ

22 at the same node (say node 2) and two bifun-

damentals X12, X21. Here W = Tr[X12[φ12, φ

22]X21] whose abelian W is again zero.

3. 2-node quiver with two bifundamentals X12,X21 and adjoints φ1 and φ2 at two dif-ferent nodes with trivial W = 0.

It is important to realise that it is not possible to do forward algorithm for quivers whoseabelian superpotential W is zero. Particularly, we cannot obtain the matrix K for thesesuperpotentials. Fortunately, the first two quivers admit dimer tiling presentation [13].

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So, we can obtain toric data G directly from the determinant of the Kastelyne matrix.We would also like to point out that the P matrix obtained from W is same for C4 andC4/Z2 indicating that the P matrix does not give information about the Chern-Simonlevels. From inverse algorithm, we actually find the P -matrix entries change with changein CS levels.

Comparing the tiling approach and the inverse algorithm, we infer that third quiveris not allowed for C4/Z2. We will see a similar situation arising in the following sectionon Fano P3.

3.2 Fano P3 theory

The symmetry group of the Calabi-Yau 4-folds constructed as complex cone over Fano P3

is SU(4) × U(1). Further b2 = 1 implies G = 2-node quiver and QD = 0. Therefore, thematrix T = G where the toric data G4×c respecting the symmetry is:

1 1 1 1 11 −1 0 0 00 1 −1 0 00 0 1 −1 0

(3.4)

Here the number of points in the toric diagram c = 5 and hence (QF )(c−4)×c will be asingle row with five entries. The first four columns in G are the external points and thelast column denotes the internal point in the toric diagram.

For the given symmetry SU(4) × U(1), we can always choose the fermionic chargeassignment QF = (aaaab) where a, b are integers. Further QF .Gt = 0 implies 4a + b = 0.Conversely, the charge QF with four entries same reflects that the Calabi-Yau 4-fold hasSU(4) symmetry. The possible choice of a, b in QF is

QF = (1, 1, 1, 1,−4) . (3.5)

The matrix K such that (K.T ) ≥ 0 turns out be again a 4× 4 matrix.1 −1 −2 −31 −1 −2 11 −1 2 11 3 2 1

So, for the toric data for P3, the K-matrix implies that the number of matter fields mustbe again 4 which is consistent with eq. (2.8).

As the toric data G for P3 is not related to toric data of the orbifold C4/Zk by GL(4,Z),we would expect that the corresponding K-matrices are not related by the GL(4,Z) andit is indeed true. Therefore, the quiver for P3 toric data has to be different from the quiverfor C4/Zk toric data.

The matching matrix P = K.T is given by:

P =

0 0 0 4 10 0 4 0 10 4 0 0 14 0 0 0 1

(3.6)

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Figure 1. Quiver Diagram for Fano P 3.

We see that the non-zero entries in the first four columns corresponding to the externalpoints of P3 toric diagram is 4. Following our results on C4/Z2, it appears that the levelsof the quiver CS theory with four matter fields will be k1 = −k2 = 4.

The quiver corresponding to P3 must be a 2-node quiver with 4 matter fields with levelsk1 = −k2 = 4, which can again have three possibilities. Two of the quivers corresponds toC4/Z4 from the tiling approach [13]. By the elimination process, we claim that the quiverdrawn in figure 1 represents the quiver gauge theory on the coincident M2 branes at thetip of the singular Calabi-Yau, which is complex cone over Fano P3 with superpotentialW = 0. All the studied quivers with NT terms in W , G nodes and E edges satisfiedNT −E+G = 0 and hence could be drawn as tiling on a two-torus [13, 15–19]. This quiversatisfies NT −E +G = −2. It will be interesting to see whether this quiver could admit a3d tiling which will help to obtain the Kastelyne matrix and the toric data [22, 23]. We alsobelieve that this quiver must be obtainable by the Higgsing of some quiver gauge theory,which does not admit tiling presentation, with three gauge group nodes.

Taking the QF (3.5) for the toric data G, we will work out the Hilbert series of themesonic moduli space. As this Fano has only one U(1) charge, which can be taken asR-charge, the Hilbert series must have the following expected form:

gmes(t;X) =1 + (g − 2)t+ (g − 2)t2 + t3

(1− t)4. (3.7)

where X denotes Fano 3-fold of genus g. Following ref. [20], we will take the R-chargefugacity of the four external points as s1 and for the internal point as 1. Then the Hilbertseries (3.7) for this case will come out to be:

gmes(s1,P3) =∮|z|=1

dz

2πiz1

(1− s1z)4(1− z−4)

=(1 + 31z4 + 31z8 + z12)

(z4 − 1)4|z=1/s1 (3.8)

Excluding the poles on the boundary of the contour, we indeed get the expected form (3.7)with the correct genus g = 33 confirming that the charge assignment and the toric data weconsidered (with no multiplicity) is correct.

We will now try to understand the inverse algorithm for other Fano 3-folds whoseb2 = 2 in the following section.

4 M2-brane theories from CY 4-folds with b2 = 2

There are four toric Fano 3-folds Bi whose second Betti number b2 = 2. They haveE = 5 external points and one internal point in the toric diagram. From the tiling/forward

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JHEP11(2011)111

algorithm, Fano B4 toric data and hilbert series were derived from a G = 3 node quivergauge theory. As (b2 − 1) = (E − 4) = (G − 2) = 1 for all these Fano Bi’s, we expect toobtain G = 3 node quivers from the inverse algorithm. Further, the number of baryonicsymmetries is (b2−1) = 1 for these Fano 3-folds. So, QD matrix will be a single row matrix.

In order to understand the pattern of QF and QD matrix, we will first do the inversealgorithm for the Fano B4 and obtain the same 3-node quiver known from the tiling/forwardalgorithm. Then, we will repeat the similar Q-charge pattern for the other three Fano 3-folds and obtain their corresponding 3-node quiver gauge theories.

4.1 Fano B4 theory

The CY 4-fold obtained from the complex cone over B4 has SU(3)×SU(2)×U(1) symmetry.The toric data G which reflects this symmetry is

G =

1 1 1 1 1 11 −1 0 0 0 00 1 −1 0 0 00 0 0 1 −1 0

. (4.1)

For the Fano with the given symmetry, the Q matrix must have first three columns areidentical, fourth and fifth column to be identical. Observing the Q-charge pattern for the14-toric Fano from the tiling/forward algorithm, we propose the following:

Ansatz: QF matrix columns must possess non-abelian symmetry whose rank is one higherthan that of the QD matrix. Further, the ranks of the non-abelian subgroups in QFmust be atmost the maximal rank of the subgroups representing the symmetry of the toricCY 4-folds.

This proposal helps in fixing the multiplicity of the points in the toric diagram as well.For B4, QF matrix must have SU(3)× SU(3) symmetry. That is, the integer entries of theQF matrix must be:

QF (c−G−2)×c =

a1 a1 a1 b1 b1 b1 c1 d1 . . .

a2 a2 a2 b2 b2 b2 c2 d2 . . .

. . . . . .

. . . . . .

ac−5 ac−5 ac−5 bc−5 bc−5 bc−5 cc−5 dc−5 . . .

, (4.2)

where the . . . denotes possible multiplicities of the points in the toric diagram. The singlerow QD entries must have SU(3)× SU(2)×U(1):

QD1×c =(x x x y y l m . . .

)(4.3)

so that the total Q possesses the symmetry of the Fano B4.For the given G with no multiplicity, we are able to find a QF and QD obeying the

above pattern:

Q =

(QFQD

)=

(1 1 1 −1 −1 −10 0 0 1 1 −2

)(4.4)

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JHEP11(2011)111

The T and K matrix for this choice of QF is:

T =

1 0 0 0 0 11 0 0 0 1 01 0 0 1 0 0−1 0 1 0 0 0−1 1 0 0 0 0

; K =

v1 v2 v3 v4 v5

X(1)12 0 0 1 0 0

X(2)12 0 0 1 0 1

X(3)12 0 0 1 1 0

X(1)23 0 1 0 0 0

X(2)23 0 1 0 0 1

X(3)23 0 1 0 1 0

X(1)31 1 0 0 0 0

X(2)31 1 0 0 0 1

X(3)31 1 0 0 1 0

(4.5)

From the K-matrix, we know that the number of matter fields is 9. We can also reconstructtoric superpotential W using the F -term constraints given by the K-matrix:

W = Tr(εijkX

(i)12X

(j)23 X

(k)31

)(4.6)

and draw the 3-node cyclic quiver where the matter fields X(l)ij ’s are bifundamental fields

from the node i to the node j which will determine the quiver charge matrix:

d =

X

(i)12 X

(j)23 X

(k)31

a = 1 1 0 −1a = 2 −1 1 0a = 3 0 −1 1

(4.7)

In order to determine the CS levels ka’s, we have to write the P = K.T matrix:

P =

p1 p2 p3 p4 p5 p6

X(1)12 1 0 0 1 0 0

X(2)12 0 1 0 1 0 0

X(3)12 0 0 1 1 0 0

X(1)23 1 0 0 0 1 0

X(2)23 0 1 0 0 1 0

X(3)23 0 0 1 0 1 0

X(1)31 1 0 0 0 0 1

X(2)31 0 1 0 0 0 1

X(3)31 0 0 1 0 0 1

, (4.8)

where we have again indicated the matter fields which represent the row index and theGLSM pα denoting the column index which will help to write these matter fields as productsof GLSM pα fields. Using the QD charge (4.4) and the P -matrix elements, we can obtainthe projected charge ∆ of the matter fields (2.6):

∆ =

(X

(1,2,3)12 X

(1,2,3)23 X

(1,2,3)31

1 1 −2

)(4.9)

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JHEP11(2011)111

Substituting the projected charge and the d-matrix elements (4.7) in eq. (2.3), we find thatthe CS levels have to be

k1 = 1 , k2 = −2 , k3 = 1 . (4.10)

Thus, using inverse algorithm with a possible choice of Q matrix respecting the ansatzon the QF , QD pattern, we have obtained the same 3-node quiver CS theory with W , CSlevels and ∆ charge [20] confirming that the inverse algorithm is agreeing with the forwardalgorithm for the Fano B4. Now, we are in a position to extend this pattern approachfor other Bi Fano 3-folds and obtain the corresponding quiver CS theories. The Hilbertseries of the mesonic moduli space worked out in ref. [20] involves taking a simple poleor irrational pole in each of the two integration variables to the boundary of the contourby scaling the variables. Evaluating the contour in the scaled variables and excluding thepoles at the boundary gives the form (3.7) agreeing with the genus g = 28.

4.2 Fano B1 theory

The symmetry possessed by this B1 Fano is SU(3)× U(1)2. Using our ansatz, we need tochoose QF to have SU(3) × SU(2) × U(1) symmetry. This could not be achieved withouttaking multiplicity of the points in the toric diagram. Hence, we start with the followingtoric data:

G =

1 1 1 1 1 1 11 −1 0 0 0 0 00 1 −1 0 0 0 00 0 2 −1 1 1 0

(4.11)

where we have taken multiplicity of an external point which is shown as repeated columns5 and 6 in G. The last column represents the internal point in the toric diagram. The QFmatrix will have now 2 rows and can be taken, following ansatz, as:

QF =

(1 1 1 −2 −2 −2 30 0 0 2 1 1 −4

)(4.12)

We can take a possible choice for the single row QD imposing its entries so that the Qmatrix respect SU(3)×U(1)2 symmetry:

QD =(

0, 0, 0, 1, 0, 1, −2)

(4.13)

With this QF , we can find T and hence the K matrix as:

T =

1 0 0 2 0 0 12 0 0 −1 0 2 02 0 0 −1 2 0 0−1 0 1 0 0 0 0−1 1 0 0 0 0 0

; K =

v1 v2 v3 v4 v5

X1 1 0 0 0 0X2 1 0 0 0 1X3 1 0 0 1 0X4 1 0 2 0 0X5 1 0 2 0 5X6 1 0 2 5 0X7 1 2 0 0 0X8 1 2 0 0 5X9 1 2 0 5 0

(4.14)

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JHEP11(2011)111

The K-matrix indicates that there are 9 matter fields Xi’s. From the K-matrix, wedo find the following relations: X9X4 = X6X7, X5X9 = X6X8, X5X7 = X4X8, andwe could construct a toric superpotential W respecting the above equation as W =Tr [X1(X5X9 −X6X8)−X2(X5X7 −X4X8) +X3(X6X7 −X9X4)] . However, the F-termconstraint ∂W

∂Xifor X4, X5, . . . X9 is not respected by the K-matrix. Hence the only possible

toric superpotential W will be a two-term superpotential with each term involving all the9 matter fields:

W = Tr [X2X5X8{X1X4X9X3X6X7 −X1X6X7X3X4X9}] . (4.15)

Clearly, abelian W = 0 and hence we cannot perform forward algorithm for this 3-nodequiver. Further NT −E+G = 2− 9 + 3 = −4 and hence cannot admit tiling presentation.

The W suggests the 3-node quiver must be as shown in figure 2. From the quiverdiagram, we can obtain the d-matrix elements:

dai =

Xi(i = 1, 2, 3) Xi(i = 4, 5, 6) Xi(i = 7, 8, 9)

a = 1 −1 1 0a = 2 0 −1 1a = 3 1 0 −1

(4.16)

To determine the CS levels of the three nodes, we need the P = K.T matrix:

P =

1 0 0 2 0 0 10 1 0 2 0 0 10 0 1 2 0 0 15 0 0 0 4 0 10 5 0 0 4 0 10 0 5 0 4 0 15 0 0 0 0 4 10 5 0 0 0 4 10 0 5 0 0 4 1

(4.17)

Using the above matrix elements and the QD (4.13), we obtain the following ∆ charge forthe nine-matter fields in the G = 3 node quiver:

∆ =

(X1,2,3 X4,5,6 X7,8,9

0 −2 2

)(4.18)

Substituting ∆ charge (4.18) and d charge (4.16) in eq. (2.3), we find that the CS levels ofthe three nodes must be

k1 = 2 , k2 = 0 , k3 = −2 . (4.19)

As a further consistent check with our charge assignment (4.12), (4.13), we will work outthe Hilbert series for the mesonic moduli space in the following subsection.

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JHEP11(2011)111

Figure 2. B1 Quiver Diagram.

4.2.1 Hilbert series for B1 theory

The total charge matrix for the B1 theory (4.12), (4.13) is

Q =

(QFQD

)=

p1 p2 p3 p4 p5 p5 p6

1 1 1 −2 −2 −2 30 0 0 2 1 1 −40 0 0 1 0 1 −2

(4.20)

Here, we have c = 7 pα fields. Let us denote the R-charge fugacity associated with thefirst three pα’s (p1, p2, p3) respecting SU(3) symmetry as s1 and fugacities associated withp4, p5 and p5 as s2, s3 and s4 respectively. Since the R-charge of the internal point p6 is0, the corresponding fugacity is set to unity. Using the Q matrix, the Hilbert series of themesonic moduli space can be written as:

gmes(s1, s2, s3, s4;B1) =∮

|z1|=1

(dz1

2πiz1

)∮|z2|=1

(dz2

2πiz2

)∮|b|=1

(db

2πib

) 1

(1− s1z1)3(

1− s2 bz22

z21

)(1− s3 z2z21

)(1− s4 z2bz21

)(1− z31

b2z42

)

Similar to the B4 integration, we have scaled each integration variable so that the simplepole (preferably irrational pole) is on the boundary. Evaluating the integration with theabove scaling, we obtain

gmes(s1, s2, s3, s4;B1) =s3s4(

s51s23s

24 − 1

)3 (s52 − s3s4

) {1 + 3s1s22 + 6s21s42

+10s31s2s3s4 + 15s41s32s3s4 + 18s51s

23s

24 + 19s61s

22s

23s

24

+18s71s42s

23s

24 + 15s81s2s

33s

34 + 10s91s

32s

33s

34 + 6s10

1 s43s

44

+3s111 s

22s

43s

44 + s12

1 s42s

43s

44

}(4.21)

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JHEP11(2011)111

Let us define two new fugacities, t1 ≡ s1/21 s

1/53 s

1/54 and t2 ≡ s2s

−1/53 s

−1/54 . So, the Hilbert

series for mesonic moduli space will be given as:

gmes(t1, t2;B1) =1(

1− t52) (

1− t101

)3 {1 + 10t2t61 + 15t2t161 + 3t22t

21 + 6t42t

41

+15t32t81 + 18t10

1 + 19t22t121 + 18t42t

141 + 10t32t

181 + 6t20

1

+3t22t221 + t42t

241

}. (4.22)

This is the expected polynomial form for the Calabi-Yau 4-folds with U(1)2 symmetry.This indirectly confirms that our charge assignment (4.20) is correct. We can determinethe R-charge assignment of the pα fields following the steps in ref. [20]. Suppose, R1 andR2 be the R-charges corresponding to the fugacities t1 and t2. That is, t1 = e−µR1 andt2 = e−µR2 , where µ is the chemical potential for the R-charge. So, the volume of B1 willbe given as:

V (B1) = limµ→0

µ4gmes(e−µR1 , e−µR2 ;B1) . (4.23)

However, R1 and R2 are not independent. Using the terms in the superpotential (4.15) tohave R-charge 2 implies that the product of all the matter fields which in terms of pα fields

from P matrix (4.17) (9∏i=1

Xi = p111 p

112 p

113 p

64p

125 p

126 p

97) must have R-charge 2 which implies

that:33R1 + 3R2 = 1 (4.24)

Putting it in volume of B1 (4.23) and minimising it, we get R1 = 0.02272 and R2 = 0.08333.The R-charge of the pα field can be found by using:

R(pα) = limµ→0

1µ

[g(e−µRi ;Dα)

gmes(e−µRi ;B1)− 1

](4.25)

where Dα is the divisor corresponding to the field pα and g(e−µRi ;Dα) gives the associatedHilbert series evaluated at the Ri which minimises V (B1). For the given charge assign-ment (4.20), the associated Hilbert series for the divisor D1 corresponding to the field p1

is

g(s1, s2, s3, s4;D1) =∮|z1|=1

(dz1

2πiz1

)∮|z2|=1

(dz2

2πiz2

)∮|b|=1

(db

2πib

)(4.26) (s1z1)−1

(1− s1z1)3(

1− s2bz22z21

)(1− s3 z2z21

)(1− s4 z2bz21

)(1− z31

b2z42

)

The SU(3) symmetry of the B1 requires R(p1) = R(p2) = R(p3). Substituting the fugacitiess1, s2, s3, s4 and rewriting in terms of t1, t2, the R-charge of p1 field turns out to be R(p1) =2R1. Similarly, computation of associated Hilbert series for divisor D4 corresponding tofield p4 gives R(p4) = R2. This method enables evaluation of R-charges of all the pα fields.In fact, R(p5) = R(p6) = 0.

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JHEP11(2011)111

SU(3) U(1)R U(1)B U(1)q fugacityp1 (1,0) 0.04545 0 1 s1y1q

p2 (1-,1) 0.04545 0 1 s1y2q/y1

p3 (0,-1) 0.04545 0 1 s1q/y2

p4 (0,0) 0.08333 1 -1 s2b/q

p5 (0,0) 0 0 -3 1/q3

p5 (0,0) 0 1 0 b

p6 (0,0) 0 -2 1 q/b2

Table 1. Various charges of pα fields under global symmetry of B1 theory. In this table,si are the fugacities of R-charges, y1 and y2 are weights of SU(3) symmetry, b and q are fugacitiesof U(1)B and U(1)q symmetries respectively.

We know that B1 has two U(1) symmetries. We have found R-charge corresponding toone U(1) symmetry. The q-charge assignment, corresponding to the other abelian mesonicsymmetry U(1), must be such that the superpotential terms (4.15) are uncharged. Furtherthe q-charge vector must be linearly independent to QF , QD (4.20). We have tabulated allthese results in table 1.

4.3 Fano B2 theory

The symmetry of B2 is also SU(3)×U(1)2. So, we expect, from anatz, QF charge assignmnetto respect SU(3)× SU(2)× U(1). In order to achieve this symmetry, we take multiplicityof two external points p4, p5 in the toric diagram giving the following toric data:

G =

p1 p2 p3 p4 p4 p5 p5 p6

1 1 1 1 1 1 1 11 −1 0 0 0 0 0 00 1 −1 0 0 0 0 00 0 1 −1 −1 1 1 0

(4.27)

and we choose the following QF charge matrix with first three columns identical, fourthand sixth column identical so that the non-abelian symmetry SU(3)× SU(2) is obeyed:

QF =

1 1 1 −1 −1 −1 −2 20 0 0 1 −1 1 −1 00 0 0 0 1 0 1 −2

(4.28)

A possible choice for the baryonic charge QD matrix with breaks the symmetry to SU(3)×U(1)2 is:

QD =(

0, 0, 0, 1, 1, 2, 0, −4)

(4.29)

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JHEP11(2011)111

Figure 3. Two quivers (1) and (2) for the B2 toric data.

From QF charge assignment (4.28), we can find the T and hence the K matrix as:

T =

2 0 0 2 2 0 0 11 0 0 0 −1 0 1 00 0 0 −1 0 1 0 0−1 0 1 0 0 0 0 0−1 1 0 0 0 0 0 0

; K =

1 0 0 0 01 0 0 0 21 0 0 2 01 0 2 0 01 0 2 0 21 0 2 2 01 2 0 0 01 2 0 0 41 2 0 4 01 2 2 0 01 2 2 0 41 2 2 4 0

(4.30)

Here, the rows denote the matter fields Xi, (i = 1, 2, . . . , 12). From the K matrix elements,we can construct a toric superpotential W as

W = Tr (X1X4X8X12 −X1X4X9X11 −X2X5X7X12

+X2X5X9X10 +X3X6X7X11 −X3X6X8X10) . (4.31)

There are NT = 6 terms in W for the G = 3 node quiver with E = 12 matter fields.Clearly, NT −E +G = −3 and cannot admit tiling presentation. However, the abelian Wis non-zero. So, it must be possible to do the forward algorithm after deducing the quiverdiagram with CS levels from the inverse algorithm. This suggests two possible quivers asshown in figure 3.

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JHEP11(2011)111

To obtain the CS levels on the three nodes of the quiver, we need the matrix P = K.T :

P =

2 0 0 2 2 0 0 10 2 0 2 2 0 0 10 0 2 2 2 0 0 12 0 0 0 2 2 0 10 2 0 0 2 2 0 10 0 2 0 2 2 0 14 0 0 2 0 0 2 10 4 0 2 0 0 2 10 0 4 2 0 0 2 14 0 0 0 0 2 2 10 4 0 0 0 2 2 10 0 4 0 0 2 2 1

Using the QD charge (4.29) and the P -matrix elements, we can obtain the projected charge∆ of the matter fields (2.6):

∆ =

(X1,2,3 X4,5,6 X7,8,9 X10,11,12

0 2 −2 0

). (4.32)

Substituting (4.32) in the charge matrix d1 for the cyclic quiver in figure 3,

d1 =

Xi(i = 1, 2, 3) Xi(i = 4, 5, 6) Xi(i = 7, 8, 9) Xi(i = 10, 11, 12)

a = 1 0 1 0 −1a = 2 0 0 −1 1a = 3 0 −1 1 0

, (4.33)

or in the charge matrix d2 of the linear quiver in figure 3

d2 =

Xi(i = 1, 2, 3) Xi(i = 4, 5, 6) Xi(i = 7, 8, 9) Xi(i = 10, 11, 12)

a = 1 1 0 0 −1a = 2 −1 1 −1 1a = 3 0 −1 1 0

, (4.34)

in eq. (2.3), the CS levels of the 3-nodes are

k1 = 2, k2 = −2, k3 = 0 . (4.35)

Incidentally, the linear and the cyclic quivers can be considered as Seiberg duals whichcorrespond to the same B2 toric data.1 To further reinforce that the charge assignmentswe have chosen is consistent, we will now do the Hilbert series of the mesonic moduli spacefor B2 toric data.

1We thank the referee for pointing this out.

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JHEP11(2011)111

4.3.1 Hilbert series evaluation for B2 theory

The total charge matrix for the B2 theory is given by:

Q =

(QFQD

)=

1 1 1 −1 −1 −1 −2 20 0 0 1 −1 1 −1 00 0 0 0 1 0 1 −20 0 0 1 1 2 0 −4

(4.36)

The symmetry group for this theory is SU(3) × U(1)2. Here, we have 8 pα fields. Letus denote the R-charge fugacity associated with the p1, p2 and p3 to be s1 and fugacitiesassociated with p4, p4, p5, p5 to be s2, s3, s4 and s5 respectively. Since the R-charge of theinternal perfect matching p6 is 0, the corresponding fugacity is set to unity. Using the Qmatrix, the Hilbert series of the mesonic moduli space can be written as:

gmes(s1, s2, s3, s4, s5;B2) =∮|z1|=1

(dz1

2πiz1

)∮|b|=1

(db

2πib

)∮|z3|=1

(dz3

2πiz3

)∮|z2|=1

(dz2

2πiz2

) 1

(1− s1z1)3(

1− s2 bz2z1)(

1− s3 bz3z1z2

)1(

1− s4 b2z2z1

)(1− s5 z3

z21z2

)(1− z21

b4z23

) (4.37)

The Hilbert series for mesonic moduli space with the change of variables as t1 = s1√s4s5

and t2 = s2s3/√s4s5 turns out to be

t23 t1

9 + 3 t18 + 6 t2 t17 + 10 t22 t16 + 12 t23 t1

5 + 12 t14 + 10 t2 t13 + 6 t22 t12 + 3 t23 t1 + 1

(t2 − 1) (t2 + 1) (t22 + 1) (t12 − 1)3 (t12 + 1)3

which is the form expected for toric CY 4-folds with two U(1) symmetries. Suppose, R1

and R2 be the R-charges corresponding to the fugacities t1 and t2. That is, t1 = e−µR1

and t2 = e−µR2 , where µ is the chemical potential for the R-charge. Similar to the volumeminimisation done for B1, we can find the value R1, R2 which minimises the volume V (B2).For the given W (4.31), we find that 3R1 + R2 = 1/2. Substituting this relation in thevolume of B2 and minimising it, we get R1 = R2 = 1/8. Following the methods of B1,using the eq. (4.36), the R-charges of each pα can be similarly determined which we tabulatein table 2.

4.3.2 Genus for B2 theory

From the fugacities of the 8 perfect matchings listed in the table 2, the Hilbert series of themesonic moduli space can be obtained by integrating over the fugacities z1, z2, z3 and b

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JHEP11(2011)111

SU(3) U(1)R U(1)B U(1)q fugacityp1 (1,0) 1/8 0 0 ty1

p2 (-1,1) 1/8 0 0 ty2/y1

p3 (0 ,-1) 1/8 0 0 t/y2

p4 (0,0) 0 1 0 b

p4 (0,0) 1/8 1 0 tb

p5 (0,0) 0 2 1 b2q

p5 (0,0) 0 0 -1 1/qp6 (0,0) 0 -4 0 1/b4

Table 2. Various charges of perfect matchings under global symmetry of B2 theory. Inthis table, t is the fugacity of R-charges, y1 and y2 are weights of SU(3) symmetry, b and q arefugacities of U(1)B and U(1)q symmetries respectively.

associated with the three rows of QF and one row of QD respectively, which is given below:

gmes(t, q, y1, y2;B2) =∮|z1|=1

(dz1

2πiz1

)∮|b|=1

(db

2πib

)∮|z3|=1

(dz3

2πiz3

)∮|z2|=1

(dz2

2πiz2

) 1

(1− ty1z1)(

1− ty2z1y1

)(1− tz1

y2

)(1− bz2

z1

)1(

1− btz3z1z2

)(1− b2qz2

z1

)(1− z3

qz21z2

)(1− z21

b4z23

) (4.38)

After doing the integration and setting the fugacities other than that of U(1)R charges as1, we get the Hilbert series as:

gmes(t, y1 = 1, y2 = 1, q = 1;B2) =1 + 31t4 + 31t8 + t12

(1− t4)4(4.39)

This result is in the form expected for the Calabi-Yau 4-folds (3.7) but it is not clear whyit is not giving the genus g = 29 [20].

4.4 Fano B3 theory

The symmetry of B3 is SU(2)2×U(1)2. We take multiplicity of internal point p5 as two inthe toric diagram giving the following toric data:

G =

p1 p2 p3 p4 p5 p5 p6

1 1 1 1 1 1 11 −1 0 0 0 0 00 0 1 −1 0 0 00 1 0 −1 −1 −1 0

(4.40)

Following ansatz, we choose the QF charge matrix with two rows respecting non-abelianSU(2)3 symmetry:

QF =

(1 1 3 3 −1 −1 −61 1 1 1 0 0 −4

)(4.41)

– 19 –

JHEP11(2011)111

Figure 4. Quiver Diagram for Fano B3.

A possible choice for the baryonic chargeQD matrix which breaks the symmetry to SU(2)2×U(1)2 is

QD =(

0, 0, 0, 0, 1, −1, 0)

(4.42)

From QF charge assignment (4.41), we can find the T and hence the K matrix as:

T =

3 0 1 0 0 0 1−1 0 1 0 0 2 0−1 0 1 0 2 0 00 0 −1 1 0 0 0−1 1 0 0 0 0 0

; K =

1 0 0 0 01 0 0 0 31 0 0 1 01 0 0 1 31 0 3 0 01 0 3 4 01 3 0 0 01 3 0 4 0

(4.43)

Here, the rows denote the matter fields Xi, (i = 1, 2, . . . , 8). From the K matrix elements,we can construct a toric superpotential W as

W = (X1X4 −X2X3)(X5X8 −X6X7) (4.44)

There are NT = 4 terms in W for the G = 3 node quiver with E = 8 matter fields. Thequiver diagram can be constructed as shown in figure 4. Clearly, NT − E + G = −1 andcannot admit tiling presentation. However, the abelian W is non-zero. So, it must bepossible to do the forward algorithm after deducing the quiver diagram with CS levelsfrom the inverse algorithm.

– 20 –

JHEP11(2011)111

To obtain the CS levels on the three nodes of the quiver, we need the matrix P = K.T :

P =

3 0 1 0 0 0 10 3 1 0 0 0 13 0 0 1 0 0 10 3 0 1 0 0 10 0 4 0 6 0 10 0 0 4 6 0 10 0 4 0 0 6 10 0 0 4 0 6 1

Using the QD charge (4.42) and the P -matrix elements, we can obtain the projected charge∆ of the matter fields and find the possible quiver as shown in figure 4 whose charge matrixd is given below:

d =

Xi(i = 1, 2) Xi(i = 3, 4) Xi(i = 5, 6) Xi(i = 7, 8)

a = 1 0 1 0 −1a = 2 0 −1 1 0a = 3 0 0 −1 1

, (4.45)

The CS levels of the 3-nodes are

k1 = 6, k2 = −6, k3 = 0 . (4.46)

4.4.1 Hilbert series evaluation for B3 theory

The total charge matrix for the B3 theory is given by:

Q =

1 1 3 3 −1 −1 −61 1 1 1 0 0 −40 0 0 0 1 −1 0

(4.47)

The symmetry group for this theory is SU(2)2 × U(1)2. Here, we have 7 pα fields. Letus denote the R-charge fugacity associated with the p1, p2 be s1, with p3, p4 be s2 andwith p5, p5 to be s3 and s4 respectively. Since the R-charge of the internal point p6 is 0,the corresponding fugacity is set to unity. Using the Q matrix, the Hilbert series of themesonic moduli space can be written as:

gmes(s1, s2, s3, s4;B3) =∮|z1|=1

(dz1

2πiz1

)∮|b|=1

(db

2πib

)∮|z2|=1

(dz2

2πiz2

) 1

(1− s1z1z2)2(1− s2z3

1z2)2 (1− s3 b

z1

)1(

1− s4 1bz1

)(1− 1

z61z42

) (4.48)

– 21 –

JHEP11(2011)111

SU(2)1 SU(2)2 U(1)R U(1)B U(1)q fugacityp1 1 0 0.164 0 1 s1x1q

p2 -1 0 0.164 0 1 s1q/x1

p3 0 1 0.782 0 -3 s2x2/q3

p4 0 -1 0.782 0 -3 s2/(x2q3)

p5 0 0 0.039 1 0 s3b

p5 0 0 0 -1 4 s4q4/b

p6 0 0 0 0 0 1

Table 3. Various charges of perfect matchings under global symmetry of B3 theory. Inthis table, si is the fugacity of R-charges, x1 and x2 are weights of two SU(2) symmetries, b and q

are fugacities of U(1)B and U(1)q symmetries respectively.

The Hilbert series for mesonic moduli space with the change of variables as t1 =(s1s

1/32

)and t2 =

(s4/32 s3s4

)turns out to be

gmes(t1, t2;B3) =1(

1− t32)2 (1− t31)3

{1 + 5t31 + 9t21t2 − 3t51t2 + 8t1t22 + t41t

22 − 3t71t

22

+3t32 − t31t32 − 8t61t32 + 3t21t

42 − 9t51t

42 − 5t41t

52 − t71t52

}. (4.49)

which is the form expected for toric CY 4-folds with two U(1) symmetries. Suppose, R1

and R2 be the R-charges corresponding to the fugacities t1 and t2. That is, t1 = e−µR1

and t2 = e−µR2 , where µ is the chemical potential for the R-charge. Similar to the volumeminimisation done for earlier cases, we can find the value R1, R2 which minimises thevolume V (B3). For the given W (4.44), we find that R1 + R2 = 1/3. Substituting thisrelation in the volume of B3 and minimising it, we get R1 = 1

24

(−3 +

√57)

= 0.189 andR2 = 1

24

(11−

√57)

= 0.144. The R-charges of each pα can be determined which wetabulate in table 3.

5 Summary and open problems

In this paper, we have systematized inverse algorithm by understanding the pattern of thecharge assignments QF , QD obtained for 14 Fano 3-folds from forward/tiling algorithm [20].Particularly, we used the second Betti number, symmetry of the CY 4-folds to fix thenumber of rows of QD charge matrix and the entries of both QF and QD matrix. Ouransatz in section 4.1, which states that the rank of the non-abelian symmetry of QF is onehigher than that of QD, indicates the multiplicity of which points in the toric data could betaken. Using the ansatz, we took a possible choice of QF , QD and performed the sequenceof steps to obtain the quiver diagram, superpotential and the Chern-Simons levels.

The quivers for Fano P3,B1,B2 and B3 as shown in figures 1–4 with the appropriate Wconstructed from K-matrix showed that they cannot admit tiling presentation. It appearsthat forward algorithm from the quiver data for B2 and B3 can be done to confirm thetoric data.

– 22 –

JHEP11(2011)111

Our charge assignments QF , QD for the four Fano 3-folds gives the expected form forthe Hilbert series of the mesonic moduli space. So, we believe that our ansatz must becorrect. Using the volume minimisation, we have tabulated the R-charge of the fields pα’s.We obtained the correct genus g = 33 for P3 Fano. However, it is not clear why the genuscomputation for B2 is giving g = 33 instead of g = 29 [20].

It will be interesting to do the higgsing approach [21] for CS quivers which does notadmit tiling presentation. We hope to report on the higgsing procedure in a future pub-lication. In ref. [24], the tiling rules for SO/Sp quivers corresponding to the orientifoldsof the CY 3-folds were proposed. It is a challenging problem to generalise the tiling fororientifolds of the CY 4-folds [25] corresponding to SO/Sp CS quivers.

Acknowledgments

We would like to thank Tapobrata Sarkar and Prabwal Phukon for discussions during theinitial stages of this project. P.R would like to thank the hospitality of Center for Quantumspacetime, Sogang University where this work was done during the sabbatical visit. Thiswork was supported by the National Research Foundation of Korea(NRF) grant funded bythe Korea government(MEST) through the Center for Quantum Spacetime(CQUeST) ofSogang University with grant number 2005-0049409.

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DOI 10.1007/s11005-013-0651-4Lett Math Phys (2013) 103:1389–1398

Multiplicity-free Quantum 6 j-Symbols for Uq(slN )

SATOSHI NAWATA1, RAMADEVI PICHAI2 and ZODINMAWIA2

1NIKHEF Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands2Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India.e-mail: [email protected]

Received: 17 April 2013 / Revised: 24 July 2013 / Accepted: 31 July 2013Published online: 21 August 2013 – © Springer Science+Business Media Dordrecht 2013

Abstract. We conjecture a closed form expression for the simplest class of multiplicity-freequantum 6 j-symbols for Uq (slN ). The expression is a natural generalization of the quan-tum 6 j-symbols for Uq (sl2) obtained by Kirillov and Reshetikhin. Our conjectured formenables computation of colored HOMFLY polynomials for various knots and links carry-ing arbitrary symmetric representations.

Mathematics Subject Classification (2010). 17B37, 17B81, 20G42, 81R50.

Keyword. quantum 6 j-symbols.

1. Introduction

From the beginning of the twentieth century, we have witnessed the mutual inter-active developments between quantum physics and representation theory. Rightfrom the birth of quantum mechanics, applications of representation theory toquantum physics have turned out to be indispensable to the study of symme-tries inherent in a quantum system. It is well-known that the Clebsch–Gordancoefficients (3 j-symbols) in decomposition of tensor product of irreducible repre-sentations of sl2(C) naturally appear in quantum theory of angular momenta. Fur-thermore, in the study of atomic spectroscopy [19], the Racah coefficients weredefined as linear combinations of products of four Clebsch–Gordan coefficients.Around the same time, by studying algebra the 3 j-symbols satisfy, Wigner inde-pendently introduced (classical) 6 j-symbols [25,26]. The 6 j-symbols are of impor-tance in all situations where the recoupling of angular momenta is involved.

Inspired by ideas coming from quantum physics, quantum deformation of uni-versal enveloping algebra of a semi-simple Lie algebra, a.k.a. a quantum group,was introduced by Drinfel’d and Jimbo [4,10], which captures the symmetry behindthe Yang–Baxter equations. Meanwhile, on a separate track, Jones [11] constructeda new polynomial invariant of links using von Neumann algebra. It soon becameapparent that the basic algebraic structures of the polynomial invariants of linkscould be described by quantum groups. This viewpoint leads to a wide class ofpolynomial invariants of links [21–23]. Furthermore, quantum groups are also

1390 SATOSHI NAWATA ET AL.

related to the two-dimensional Wess–Zumino–Novikov–Witten (WZNW) model.The actions of braid groups on conformal blocks of WZNW model turn out to beequivalent to the braid group representation obtained from the universal R-matrixof the quantum group [1,2,5,14]. Hence, the polynomials invariants of links canbe obtained from monodromy representations along solutions of the Knizhnik–Zamolodchikov equations.

It has been proven that the quantum deformations of the Clebsch–Gordan coef-ficients and the 6 j-symbols which naturally appear in the representation theory ofquantum groups connect these areas of mathematics and physics in a beautiful way[13]. However, the generalization of the quantum 6 j-symbols for Uq(sl2) [13] tohigher ranks has remained a challenging open problem. In this paper, we shall con-jecture a closed form expression of the simplest class of the quantum 6 j-symbolsfor Uq(slN ).

2. Quantum 6 j -Symbols

Let us denote the spin- j representation of Uq(sl2) by Vj whose highest weight isλ = 2 j ( j ∈ 1

2 Z). The space of four-point conformal blocks is the space of linearmaps HomUq (sl2)(Vj1 ⊗ Vj2 ⊗ Vj3 ⊗ V ∗

j4,C) invariant under the diagonal action of

Uq(sl2) and, most importantly, it is a finite-dimensional vector space. By associa-tivity of the tensor product, we can take a basis in two ways;

(1)

and

(2)

where j12 and j23 satisfy the quantum Clebsch–Gordan condition (the fusion rule);j1 + j2 + j12 ∈ Z and | j1 − j2| ≤ j12 ≤ j1 + j2. (The same condition for j23.) Then,

the quantum 6 j-symbols{

j1 j2 j12

j3 j4 j23

}for Uq(sl2) appear in the transformation

matrix of the two bases

(3)

where

a j12 j23

[j1 j2j3 j4

]= (−1) j1+ j2+ j3+ j4

√[2 j12 +1][2 j23 +1]{

j1 j2 j12

j3 j4 j23

}. (4)

MULTIPLICITY-FREE QUANTUM 6J -SYMBOLS 1391

Here, the square bracket defines a q-number

[n]= qn/2 −q−n/2

q1/2 −q−1/2,

and the transformation matrix a j12 j23 is usually called the fusion matrix. The rigor-ous derivation of a closed form expression of the quantum 6 j-symbols for Uq(sl2)

was first given by Kirillov and Reshetikhin [13]. Later, Masbaum and Vogel pro-vided another derivation based on linear skein theory [15].

The generalization to higher ranks requires replacement of a spin j by a high-est weight λ of a representation of Uq(slN ). A representation of Uq(slN ) with ahighest weight λ can be equivalently specified by a Young tableau {�i }1≤i≤N−1

with �1 ≥ · · · ≥ �N−1. If one writes the highest weight as λ = ∑N−1i=1 λiωi where

{ωi }1≤i≤N−1 are the fundamental weights, the relation to the Young tableau canbe read off by �i = λi + λi+1 + · · · + λN−1. In what follows, we identify a highestweight with a Young tableau by this dictionary. For general N , it is necessary tointroduce the conjugate representation Vλ∗ of the representation with the highestweight λ where λ∗ :=∑N−1

i=1 λN−iωi . Then, the fusion rule of quantum 6 j-symbolsfor Uq(slN )

{λ1 λ2 λ12

λ3 λ4 λ23

}(5)

is that Vλ12 ∈ (Vλ1 ⊗ Vλ2)∩ (Vλ∗3⊗ Vλ∗

4) and Vλ23 ∈ (Vλ2 ⊗ Vλ3)∩ (Vλ∗

1⊗ Vλ∗

4). From the

construction, we expect the quantum 6 j-symbols to satisfy the following symme-tries: {

λ1 λ2 λ12

λ3 λ4 λ23

}=

{λ3 λ2 λ23

λ1 λ4 λ12

}=

{λ∗

1 λ∗2 λ∗

12λ∗

3 λ∗4 λ∗

23

}

={λ1 λ4 λ∗

23λ3 λ2 λ∗

12

}=

{λ2 λ1 λ12

λ4 λ3 λ∗23

}(6)

In addition, the relationship between the fusion matrix and the quantum6 j-symbols for Uq(slN ) is generalized to

aλ12λ23

[λ1 λ2

λ3 λ4

]= ε{λi }

√dimq Vλ12 dimq Vλ23

{λ1 λ2 λ12

λ3 λ4 λ23

}, (7)

where ε{λi } = ±1 and dimq Vλ is the quantum dimension of the representation Vλ

with highest weight λ.Unlike the representations of Uq(sl2), there are serious technical difficulties for

Uq(slN ) in the fact that the decompositions of tensor products involve multiplic-ity structure in general. Specifically, isomorphic irreducible constituents will arisemore than once in the decomposition of a tensor product. However, there are spe-cial cases which decompose in a multiplicity-free way: the tensor products of twosymmetric representations, and the tensor products of a symmetric representation


and a representation conjugate to a symmetric representation. In this paper, werestrict ourselves to consider such cases for brevity. To obey the fusion rule, twoof λ1, λ2, λ3, λ4 need to be symmetric representations, and the other two must beconjugate to symmetric representations. Using the symmetries (6), it turns out thatmultiplicity-free quantum 6 j-symbols for Uq(slN ) amount to the following twotypes:

• Type I

(8)

where n2 ≤ n1 ≤ n3, k1 ≤ n2 and k2 ≤ n1. The fusion rule requires n1 + n3 =n2 +n4.

• Type II

(9)

where n1 ≤n2, k2 ≤min(n1,n3) and k1 ≤min(n1,n3,n4). The fusion rule requiresn1 +n2 =n3 +n4.

We note that the highest weight of the symmetric representation is nω1

and that of the conjugate representation is nωN−1 where represents

(N −1) vertical boxes .For low representations (ni ≤ 2), we have determined the quantum 6 j-symbols

[20,27]. Extension to higher representations (ni ≤ 4) is achieved through the fol-lowing route: In our recent paper [17], we have colored HOMFLY polynomialsfor a class of knots called twist knots K p where p denotes the number of full-twists. Using the identities obeyed by quantum 6 j-symbols and equating the col-ored HOMFLY polynomials of these twist knots with the SU (N ) Chern–Simonsinvariant involving quantum 6 j-symbols (see (C.1) in [17]), we determined thequantum 6 j-symbols for ni =3 and ni =4. We recognized a pattern from the dataon quantum 6 j-symbols (ni ≤4) motivating us to attempt a closed form expressionfor arbitrary ni ’s.


CONJECTURE. The closed form expression for quantum 6 j-symbols of type Iand type II is as follows:

{λ1 λ2 λ12

λ3 λ4 λ23

}=�(λ1, λ2, λ12)�(λ3, λ4, λ12)�(λ1, λ4, λ23)�(λ2, λ3, λ23)

×[N −1]!∑

z∈Z≥0

(−)z[z + N −1]!Cz({λi }, λ12, λ23)

×{[

z − 12〈λ1 +λ2 +λ12, α

∨1 +α∨

N−1〉]!

×[

z − 12〈λ3 +λ4 +λ12, α

∨1 +α∨

N−1〉]!

×[

z − 12〈λ1 +λ4 +λ23, α

∨1 +α∨

N−1〉]!

×[

z − 12〈λ2 +λ3 +λ23, α

∨1 +α∨

N−1〉]!

×[

12〈λ1 +λ2 +λ3 +λ4, α

∨1 +α∨

N−1〉− z

]!

×[

12〈λ1 +λ3 +λ12 +λ23, α

∨1 +α∨

N−1〉− z

]!

×[

12〈λ2 +λ4 +λ12 +λ23, α

∨1 +α∨

N−1〉− z

]!}−1

, (10)

where

�(λ1, λ2, λ3)={[

12〈−λ1 +λ2 +λ3, α

∨1 +α∨

N−1〉]!

×[

12〈λ1 −λ2 +λ3, α

∨1 +α∨

N−1〉]!

×[

12〈λ1 +λ2 −λ3, α

∨1 +α∨

N−1〉]!}1/2

×{[

12〈λ1 +λ2 +λ3, α

∨1 +α∨

N−1〉+ N −1]!}−1/2

. (11)

Here, we define the q-factorial by [n]!= [n][n −1] · · · [3][2][1] and take [0]!=1. Weuse the fact that α∨

j are duals of simple roots which form a basis of coroots, andthe paring with the fundamental weights provides 〈ωi , α

∨j 〉= δi j .

It is important to stress that only finite number of positive integers z in the sum-mation will give non-zero contribution. The factors Cz({λi }, λ12, λ23) in (10) fortype I are


Cz

({n1ω1 n2ωN−1 (n1 −n2 + k1)ω1 + k1ωN−1

n3ω1 n4ωN−1 (n3 −n2 + k2)ω1 + k2ωN−1

})

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

δz,zmin+i

[N −2+ k2 − i

k2 − i

]−1

for k1 > k2 ,

δz,zmin+i

[N −2+ k1 − i

k1 − i

]−1

for k1 ≤ k2 ,

(12)

where zmin is the smallest integer z in the summation in (10) which gives a non-trivial value. Similarly, for type II, the factors Cz({λi }, λ12, λ23) are

Cz

({n1ω1 n2ω1 (n1 +n2 −2k1)ω1 + k1ω2

n3ωN−1 n4ωN−1 (n2 −n3 + k2)ω1 + k2ωN−1

})

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

δz,zmax−i

[N −2+ k2 − i

k2 − i

]−1

for k1 > k2 ,

δz,zmax−i

[N −2+ k1 − i

k1 − i

]−1

for k1 ≤ k2 ,

(13)

where zmax is the largest integer z in the summation in (10) which gives a non-trivial value. We denote the q-binomial by[

pq

]= [p]!

[q]![p −q]! . (14)

Although there are the square roots in the expression (11), the 6 j-symbols (10)are actually rational functions with respect to q1/2. Obviously, it is easy to seethat the expression (10) reduces to the form of Uq(sl2) provided by Kirillov andReshetikhin [13] when we take N =2.

In addition, we have checked that (10) satisfy the orthogonal property

∑λ12

dimq Vλ12 dimq Vλ23

{λ1 λ2 λ12

λ3 λ4 λ23

}{λ1 λ2 λ12

λ3 λ4 λ′23

}= δλ23λ

′23

, (15)

and the Racah identity

∑λ12

ε{λ12,λ23,λ24}q− C122 dimq Vλ12

{λ1 λ2 λ12

λ3 λ4 λ23

}{λ1 λ2 λ12

λ4 λ3 λ24

}

={λ3 λ2 λ23

λ4 λ1 λ24

}q

C23+C242 q− C1+C2+C3+C4

2 , (16)

as well as the identity{λ1 λ2 0λ3 λ4 λ23

}= ε{λ2,λ3,λ23}δλ1λ2δλ3λ4√

dimq Vλ2 dimq Vλ3

. (17)

Here Ci is the quadratic Casimir invariant for the representation of highest weightλi . The sign ε = ±1 can be easily read off by comparing it with the results forUq(sl2).


We expect that the quantum 6 j-symbols for Uq(slN ) obey the pentagon (Bieden-harn-Elliot) identity

∑μ1

ε{μi ,κi } dimq Vμ1

{κ1 λ3 κ2

λ4 λ5 μ1

} {λ1 λ2 κ1

μ1 λ5 μ2

} {μ2 λ2 μ1

λ3 λ4 μ3

}

= ε{λi }{λ1 μ3 κ2

λ4 λ5 μ2

} {λ1 λ2 κ1

λ3 κ2 μ3

}. (18)

However, this property cannot be checked unless expressions beyond the multiplicity-free ones are obtained.

With our conjectured quantum 6 j-symbols (10), we can verify that they repro-duce the HOMFLY polynomials colored by the symmetric representation of manynon-torus knots [9,8], the Whitehead link and the Borromean rings [12,7] up to 4boxes. Moreover, we compute colored HOMFLY polynomials of many knots andlinks which we tabulate in the companion paper [16]. However, it is not clear, atpresent, whether the proof [13,15] can be extended to our conjectured slN quan-tum 6 j-symbols mainly due to the presence of Cz({λi }, λ12, λ23) in (10).

3. Discussion

In this paper, we proposed the closed form expression of the quantum 6 j-symbolsfor Uq(slN ). However, the structure behind 6 j-symbols is by far richer and we onlyscratch the surface of this topic. Firstly, a rigorous derivation of (10) by quantumgroups still remains an open problem. In addition to this, further study needs tobe undertaken to obtain the expressions for arbitrary representations. For this pur-pose, it is necessary to investigate explicit expressions for the quantum 3 j-symbolsand their relation to the quantum 6 j-symbols.

There is an important property of quantum 6 j-symbols for Uq(slN ) which willbe useful for evaluating quantum 6 j-symbols beyond symmetric representations.We can observe that a quantum 6 j-symbol involving anti-symmetric representa-tions and their conjugate representations can be obtained by changing [N + k]→[N − k] in the quantum 6 j-symbol for symmetric representations related by trans-position (mirror reflection across the diagonal). For example, we can find the fol-lowing 6 j-symbols using properties (15), (16) and (17):

(19)

The quantum 6 j-symbol related by transposition is

(20)


It is easy to see the symmetry [N + k] ↔ [N − k] between the two quantum6 j-symbols. In fact, this explains the symmetry between the colored HOMFLYpolynomials colored by symmetric representation Sr and anti-symmetric represen-tations r :

PSr (K ;a,q)= Pr (K ;a,q−1). (21)

Following [6], we call this property the mirror symmetry of quantum 6 j-symbols.In this way, every multiplicity-free quantum 6 j-symbol involving anti-symmetricrepresentations can be obtained from its mirror dual 6 j-symbol which can beexplicitly evaluated by (10). Nevertheless, a closed form expression using highestweights as in (10) is still lacking in anti-symmetric representations. Certainly, itwould be intriguing to see whether the mirror symmetry holds beyond multiplicity-free quantum 6 j-symbols.

Another important aspect of quantum 6 j-symbols is their relationship withq-hypergeometric functions [3]. The quantum 6 j-symbols for Uq(sl2) can beexpressed as the balanced hypergeometric function 4φ3 [13]. However, for Uq(slN ),the coefficients Cz({λi }, λ12, λ23) prevents us from writing the expression (10) interms of the balanced hypergeometric function 4φ3. Therefore, it is important tostudy the connection to generalized q-hypergeometric functions [3]. Besides, it iswell-known that there are many different ways to express the quantum 6 j-symbolsfor Uq(sl2). Hence, it would be worthwhile to find the other expressions for thequantum 6 j-symbols for Uq(slN ).

Furthermore, there is a geometric interpretation of the quantum 6 j-symbols.One can associate the quantum 6 j-symbols to a tetrahedron whose edges are col-ored by representations of Uq(slN ) [24]. Although it is necessary to have quantum6 j-symbols for arbitrary representations to obtain invariants of 3-manifolds [24]in the context of Uq(slN ), the expression (10) is suitable to study the large colorbehavior [18]. Therefore, it would be interesting to explore the large color behaviorof quantum 6 j-symbols and their relation to the geometry of the complement ofa tetrahedron in S3.

As we have seen, quantum 6 j-symbols are very interesting in their own rightand contain remarkable mathematical structure. Despite their long history, they areindeed among the least understood quantities in mathematical physics. While inthis paper we focus on multiplicity-free quantum 6 j-symbols for Uq(slN ), we hopethat our results will serve as a stepping stone towards the study of general quan-tum 6 j-symbols.

Acknowledgements

The authors are indebted to Stavros Garoufalidis and Jasper Stokman for theuseful discussions and comments. S.N. and Z. would like to thank Indian StringMeeting 2012 at Puri for providing a stimulating academic environment. S.N. isgrateful to IIT Bombay for its warm hospitality. The work of S.N. is partially sup-


ported by the ERC Advanced Grant no. 246974, “Supersymmetry: a window tonon-perturbative physics”.

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