diapositive 1title: diapositive 1 author: nabil bouchair created date: 6/25/2014 11:37:26 am
TRANSCRIPT
![Page 1: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/1.jpg)
String Amplitudes, Topological Strings and the Omega-deformation
Ahmad Zein Assi
CERN
Strings @ Princeton
26 - 06 - 2014
![Page 2: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/2.jpg)
String Amplitudes, Topological Strings and the Omega-deformation
Ahmad Zein Assi
CERN
Strings @ Princeton
26 - 06 - 2014
Based on work with
I. AntoniadisI. FlorakisS. HoheneggerK. S. Narain
1302.6993 [hep-th]
1309.6688 [hep-th]
1406.xxxx [hep-th]
![Page 3: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/3.jpg)
Introduction & Motivations
Topological String: subsector of String Theory– Twisted version of type II
– Free energy Fg = physical coupling <(R-)2 (T-)
2g-2>
Witten (88’)
Antoniadis, Gava, Narain, Taylor (93’)
Bershadsky, Ceccotti, Ooguri, Vafa (93’)
![Page 4: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/4.jpg)
Introduction & Motivations
Topological String: subsector of String Theory– Twisted version of type II
– Free energy Fg = physical coupling <(R-)2 (T-)
2g-2>
Geometric engineering of (Ω-deformed) supersymmetric gauge theories
Witten (88’)
Antoniadis, Gava, Narain, Taylor (93’)
Bershadsky, Ceccotti, Ooguri, Vafa (93’)
Nekrasov et al. (02’)
Katz, Klemm, Vafa (96’)
![Page 5: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/5.jpg)
Introduction & Motivations
Topological String: subsector of String Theory– Twisted version of type II
– Free energy Fg = physical coupling <(R-)2 (T-)
2g-2>
Geometric engineering of (Ω-deformed) supersymmetric gauge theories
Refinement: one-parameter extension of Fg
– Does it exist?
Coupling in the string effective action?
Witten (88’)
Antoniadis, Gava, Narain, Taylor (93’)
Bershadsky, Ceccotti, Ooguri, Vafa (93’)
Nekrasov et al. (02’)
Katz, Klemm, Vafa (96’)
![Page 6: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/6.jpg)
Answer from the string effective action
Ω-background– ε- ↔ SU(2)- rotation, ε+ ↔ SU(2)+ rotation
– T- → anti-self-dual graviphoton field strength
– F+ → self-dual gauge field strength
![Page 7: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/7.jpg)
Answer from the string effective action
Ω-background– ε- ↔ SU(2)- rotation, ε+ ↔ SU(2)+ rotation
– T- → anti-self-dual graviphoton field strength
– F+ → self-dual gauge field strength
Consider Fg,n = <(R-)2(T-)
2g-2(F+)2n>– Heterotic on K3 x T2 : vector partner of T2 Kähler modulus
– Contributions start at one-loop
![Page 8: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/8.jpg)
Answer from the string effective action
Ω-background– ε- ↔ SU(2)- rotation, ε+ ↔ SU(2)+ rotation
– T- → anti-self-dual graviphoton field strength
– F+ → self-dual gauge field strength
Consider Fg,n = <(R-)2(T-)
2g-2(F+)2n>– Heterotic on K3 x T2 : vector partner of T2 Kähler modulus
– Contributions start at one-loop
Explicit exact evaluation at one-loop in Heterotic
![Page 9: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/9.jpg)
Answer from the string effective action
Ω-background– ε- ↔ SU(2)- rotation, ε+ ↔ SU(2)+ rotation
– T- → anti-self-dual graviphoton field strength
– F+ → self-dual gauge field strength
Consider Fg,n = <(R-)2(T-)
2g-2(F+)2n>– Heterotic on K3 x T2 : vector partner of T2 Kähler modulus
– Contributions start at one-loop
Explicit exact evaluation at one-loop in Heterotic
![Page 10: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/10.jpg)
Answer from the string effective action
Non-perturbative corrections– Instantons in the Ω-background: deformed ADHM
– Gauge instantons: Dp-Dp+4 configuration
– Closed string background: T- and F+
![Page 11: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/11.jpg)
Answer from the string effective action
Non-perturbative corrections– Instantons in the Ω-background: deformed ADHM
– Gauge instantons: Dp-Dp+4 configuration
– Closed string background: T- and F+
Compute Ω-deformed ADHM action– Take α’ → 0 limit
![Page 12: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/12.jpg)
Answer from the string effective action
Non-perturbative corrections– Instantons in the Ω-background: deformed ADHM
– Gauge instantons: Dp-Dp+4 configuration
– Closed string background: T- and F+
Compute Ω-deformed ADHM action– Take α’ → 0 limit
Evaluate the instanton path-integral
![Page 13: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/13.jpg)
Answer from the string effective action
Non-perturbative corrections– Instantons in the Ω-background: deformed ADHM
– Gauge instantons: Dp-Dp+4 configuration
– Closed string background: T- and F+
Compute Ω-deformed ADHM action– Take α’ → 0 limit
Evaluate the instanton path-integral
![Page 14: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/14.jpg)
Holomorphicity Properties of the Refined Couplings
Explicit breaking of holomorphicity– Related to the compactness of the CY
![Page 15: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/15.jpg)
Holomorphicity Properties of the Refined Couplings
Explicit breaking of holomorphicity– Related to the compactness of the CY
Non-compact CY for– R-symmetry current
– Decoupling of hypers
– Define generating function refined topological invariants
Huang, Kashani-Poor, Klemm, Vafa, etc.
![Page 16: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/16.jpg)
Holomorphicity Properties of the Refined Couplings
Explicit breaking of holomorphicity– Related to the compactness of the CY
Non-compact CY for– R-symmetry current
– Decoupling of hypers
– Define generating function refined topological invariants
Use the generic CY compactification + appropriate limit
Huang, Kashani-Poor, Klemm, Vafa, etc.
![Page 17: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/17.jpg)
Holomorphicity Properties of the Refined Couplings
Explicit breaking of holomorphicity– Related to the compactness of the CY
Non-compact CY for– R-symmetry current
– Decoupling of hypers
– Define generating function refined topological invariants
Use the generic CY compactification + appropriate limit
Fg,n satisfies a generalised holomorphic anomaly equation (à la Klemm, Walcher, etc.)
Huang, Kashani-Poor, Klemm, Vafa, etc.
![Page 18: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/18.jpg)
Accident? Coincidence?
Background of anti-self-dual graviphotons + self-dual T-vectors = consistent string theory uplift of the Ω-background– Perturbative Nerkrasov partition function = one-loop
effective action of generalized F-terms (in the field theory limit)
– Non-perturbative part = tree-level effective action of a Dp-D(p+4) bound state
![Page 19: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/19.jpg)
Accident? Coincidence?
Background of anti-self-dual graviphotons + self-dual T-vectors = consistent string theory uplift of the Ω-background– Perturbative Nerkrasov partition function = one-loop
effective action of generalized F-terms (in the field theory limit)
– Non-perturbative part = tree-level effective action of a Dp-D(p+4) bound state
Generalised holomorphic anomaly equations
![Page 20: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/20.jpg)
Accident? Coincidence?
Background of anti-self-dual graviphotons + self-dual T-vectors = consistent string theory uplift of the Ω-background– Perturbative Nerkrasov partition function = one-loop
effective action of generalized F-terms (in the field theory limit)
– Non-perturbative part = tree-level effective action of a Dp-D(p+4) bound state
Generalised holomorphic anomaly equations
Promising candidate for a worldsheet realization of the refined topological string
![Page 21: Diapositive 1Title: Diapositive 1 Author: Nabil Bouchair Created Date: 6/25/2014 11:37:26 AM](https://reader036.vdocuments.site/reader036/viewer/2022071219/6057ee0e6afb1a63d046a8d5/html5/thumbnails/21.jpg)
String Amplitudes, Topological Strings and the Omega-deformation
Ahmad Zein Assi
CERN
Strings @ Princeton
26 - 06 - 2014
Thank You!