Numerical studies of the ABJM theoryfor arbitrary N at arbitrary coupling constant

Masazumi HondaSOKENDAI & KEK

In collaboration with Masanori Hanada (KEK), Yoshinori Honma (SOKENDAI & KEK),Jun Nishimura (SOKENDAI & KEK), Shotaro Shiba (KEK) &Yutaka Yoshida (KEK)

Reference: JHEP 0312 164(2012) (arXiv:1202.5300 [hep-th])

名古屋大弦理論セミナー　２０１２年４月２３日

IntroductionF

(free energy)

N3/2Surprisingly,

we can realize this result even by our laptop

AdS4/CFT3

Numerical simulation of U(N)×U(N) ABJM on S3

Motivation ： ( k: Chern-Simons level )

[Aharony-Bergman-Jafferis-Maldacena ’08]

ＡＢＪＭ theory

　　 relatively easy

relatively hard

（ Intermediate ）Extremely difficult ！Key for a relation between string and M-theory?

Investigate the whole region by numerical simulation!

This talk is about…

4

Monte Carlo calculation of the Free energy in U(N)×U(N) ABJM theory on S3 （ with keeping all symmetry ）

・ Test all known analytical results

・ Relation between the known results and our simulation result

Developments on ＡＢＪＭ Free energy・ June 2008 ： ABJM was born.

[Drukker-Marino-Putrov]

[Herzog-Klebanov-Pufu-Tesileanu]

Agrees with SUGRA’s result!!

Formally same( ※ λ=N/k)

・ July 2010 ： Planar limit for strong coupling

・ November 2010: Calculation for k=fixed, N→∞

[Cf. Cagnazzo-Sorokin-Wulff ’09]

[Aharony-Bergman-Jafferis-Maldacena]

～ string wrapped on CP1 CP⊂ 3 ＝ worldsheet instanton ?

※CP ３ has nontrivial 2-cycle

（ Cont’d ） Development on ABJM free energy・ June 2011 : Summing up all genus around planar limit for strong λ

[Okuyama]

[Marino-Putrov]

[Fuji-Hirano-Moriyama]

where

・ October 2011 ： Calculation for k<<1, k<<N

・ October 2011 ： Exact calculation for N=2 Formally same

Correction to Airy function→How about for large k??・ February 2012 ： Numerical simulation in the whole region(=this talk ）

At least up to instanton effect, for all k,

Free energy is a smooth function of k !![ Hanada-M.H.-Honma-Nishimura-Shiba-Yoshida]

Contents

7

1. Introduction & Motivation2. How to put ABJM on a computer3. Result4. Interpretation5. Summary & Outlook

How do we ABJM on a computer?

8

Action ：～ Approach by the orthodox method （ =Lattice ）～

・ It is not easy to construct CS term on a lattice・ It is generally difficult to treat SUSY on a lattice

Difficulties in “formulation”

Practical difficulties ・∃ Many fermionic degrees of freedom → Heavy computational costs・ CS term = purely imaginary → sign problem

[Cf. Bietenholz-Nishimura ’00]

[Cf. Giedt ’09]

hopeless…

(Cont’d)How do we put ABJM on a computer?

9

We can apply the localization method for the ABJM partition function

Lattice approach is hopeless…

Localization method[Cf. Pestun ’08]Original partition function:

where

1 parameter deformation:

Consider t-derivative:

Assuming Q is unbroken

We can use saddle point method!!

(Cont’d) Localization method

11

Consider fluctuation around saddle points:

where

Localization of ABJM theory[Kapustin-Willet-Yaakov ’09]

(Cont’d) Localization of ABJM theory

Saddle point:

Matter 1-loop

Gauge 1-loop

CS term

14

After applying the localization method,

Sign problem

(Cont’d)How do we put ABJM on a computer?

the partition function becomes just 2N-dimentional integration:

Further simplification occurs!!

Simplification of ABJM matrix model

15

[Kapustin-Willett-Yaakov ’10, Okuyama ‘11, Marino-Putrov ‘11]

Cauchy identity:

Fourier trans.:

16

(Cont’d) Simplification of ABJM matrix model

Gaussian integration

Fourier trans.:

Cauchy id.:

Short summary

17

Cauchy identity, Fourier trans. & Gauss integration

Complex≠probability

Lattice approach is hopeless… ( SUSY, sign problem, etc)∵Localization method

Easy to perform simulation even by our laptop

How to calculate the free energyProblem: Monte Carlo can calculate only expectation value

We regard the partition function as an expectation value under another ensemble:

VEV under the action:Note:

Contents

19

1. Introduction & Motivation2. How to put ABJM on a computer3. Result4. Interpretation5. Summary & Outlook

Warming up: Free energy for N=2[ Okuyama ’11]

for odd k

for even k

k

F(free energy)

( CS level )

There is the exact result for N=2:

Complete agreement with the exact result !!

Result for Planar limit [Drukker-Marino-Putrov ’10]

・ Weak couling ：・ Strong coupling ：

Weak coupling

Strong coupling

strong weak strong weak

Different from worldsheet instanton behavior

Worldsheet instanton

3/2 power low in 11d SUGRA limit

F

N3/2

11d classical SUGRA:

1/N

F/N3/2

[Drukker-Marino-Putrov ‘10, Herzog-Klebanov-Pufu-Tesileanu ‘10]

(Cont’d) 3/2 power low in 11d SUGRA limit

11d classical SUGRA:

Perfect agreement !!

Comparison with Fuji-Hirano-Moriyama

strong weak

Weak coupling

FHM

Almost agrees with FHM for strong coupling→more precise comparison by taking difference

Ex.) For N=4

Almost agrees with FHM for strong coupling→more precise comparison by taking difference

Discrepancy independent of Ｎ and dependent on ｋ→different from instanton bahavior ( ～ exp dumped)

[Fuji-Hirano-Moriyama ’11]

Contents

25

1. Introduction & Motivation2. How to put ABJM on a computer3. Result4. Interpretation5. Summary & Outlook

Fermi gas approach[Marino-Putrov ’11]

Cauchy id.:

where

Our result says that this remains even for large k??

Regard as a Fermi gas system

Result:

Origin of Discrepancy for the Planar limit (without MC)

27

[Marino-Putrov ‘10]

Analytic continuation:

Lens space L(2,1)=S3/Z2 matrix model:[Cf. Yost ’91, Dijkgraaf-Vafa ‘03]

Genus expansion:

(Cont’d))Origin of Discrepancy for the Planar limit (without MC)[Drukker-Marino-Putrov ‘10]

The “derivative” of planar free energy is exactly found as

By using asymptotic behavior,

We impose the boundary condition: Cf.

Necessary for satisfying b.c. , taken as 0 for previous works

Origin of discrepancy for all genus

29

This is explained by ``constant map’’ contribution in language of topological string:

Discrepancy is fitted by

Divergent, but Borel summable:

[ Bershadsky-Cecotti-Ooguri-Vafa ’93, Faber-Pandharipande ’98, Marino-Pasquwtti-Putrov ’09 ]

Comparison with discrepancy and Fermi gas

Fermi Gas

genus 2

Divergent, but Borel summable:

Borel sum of Constant map realizesFermi Gas （ small k ） result ！！→Can we understand the relation analytically?

Fermi Gas from Constant map

31

Borel

Expand around k=0

　 Agrees with Fermi Gas result ！

Constant map contribution ：

True for all k?

All order form ？

→Fermi Gas result is asymptotic series around k=0

Contents

32

1. Introduction & Motivation2. How to put ABJM on a computer3. Result4. Interpretation5. Summary & Outlook

Summary

・ The free energy for whole region up to instanton effect ：

～ instanton effect

・ Predict all order form of Fermi Gas result ：where

・ Discrepancy from Fuji-Hirano-Moriyama not originated by instantons　 is explained by constant map contribution

・ Although summing up all genus constant map is asymptotic series, it is Borel summable.

Monte Carlo calculation of the Free energy in U(N)×U(N) ABJM theory on S3 （ with keeping all symmetry ）

Problem

34

・ What is a physical meaning of constant map contribution?

- In Fermi gas description, this is total energy of membrane instanton[ Becker-Becker-Strominger ‘95]

- Why is ABJM related to the topological string theory?

・ Mismatch between renormalization of ‘t Hooft coupling and AdS radius[ Bergmanr-Hirano ’09]

・ If there is also constant map contribution on the gravity side, there are α’-corrections at every order of genus

[ Kallosh-Rajaraman ’98]- Does it contradict with the proof for non-α’-correction? - Is constant map origin of free energy on the gravity side??

Outlook

35

Monte Carlo method is very useful to analyze unsolved matrix models.In particular, there are many interesting problemsfor matrix models obtained by the localization method.Example(3d):・ Other observables Ex.) BPS Wilson loop・ Other gauge group ・ On other manifolds Ex.) Lens space・ Other theory Ex.) ABJ theory ・ Nontrivial test of 3d duality・ Nontrivial test of F-theorem for finite N

[ Hanada-M.H.-Honma-Nishimura-Shiba-Yoshida, work in progress]

Example(4d):・

[ M.H.-Imamura-Yokoyama, work in progress]

[ Azeyanagi-Hanada-M.H.-Shiba, work in progress]

Example(5d):・

[ M.H.-Honma-Yoshida, work in progress]

完36

Appendix

37

“Direct” Monte Carlo method(≠Ours)

38

Ex.) The area of the circle with the radius 1/2

① Distribute random numbers many times

② Count the number of points which satisfy

③ Estimate the ratio

.. ..

... .

.. . .

..

.

..

..

. .. . .

..

. .

..

..

..

...

.. .

.

.. .

..

.. .

..

..

....

.

.

..

..

.

.

. ..

..

..

Note: This method is available only for integral over compact region

“Markov chain” Monte Carlo (=Ours)

39

Ex.) Gaussian ensemble (by heat bath algorithm)

① Generate random configurations with Gaussian weight many times

② Measure observable and take its average

We can generate the following Markov chain from the uniform random numbers:

Essence of Markov chain Monte Carlo

40

Consider the following Markov process:

“sweep”

Under some conditions,

“thermalization”

transition prob. monotonically converges to an equilibrium prob.

We need an algorithm which generates

“Hybrid Monte Carlo algorithm” is useful !!

Hybrid Monte Carlo algorithm[ Duane-Kennedy-Pendleton-Roweth ’87]

(Detail is omitted. Please refer to appendix later.)

① Take an initial condition freely② Generate the momentum with Gaussian weight

Regard as the “conjugate momentum”

③ Solve “Molecular dynamics”

④ Metropolis test

accepted

“Hamiltonian”:

accepted with prob.

rejected with prob.

[ Cf. Rothe, Aoki’s textbook]

Note on Statistical Error

42

If all configurations were independent of each other,

Average:

However,

all configurations are correlated with each other generally.

Error analysis including such a correlation = “Jackknife method”

(file: jack_ABJMf.f , I omit the explanataion. )

Taking planar limit

N=8

44

Higher genus

45