Numerical studies of the ABJM theory for arbitrary

Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant Masazumi Honda SOKENDAI & KEK Reference: JHEP 0312 164(2012) (ar. Xiv: 1202. 5300 [hep-th]) In collaboration with Masanori Hanada (KEK), Yoshinori Honma (SOKENDAI & KEK), Jun Nishimura (SOKENDAI & KEK), Shotaro Shiba (KEK) & Yutaka Yoshida (KEK) 名古屋大弦理論セミナー　２０１２年４月２３日

F Introduction (free energy) Surprisingly, N 3/2 we can realize this result even by our laptop

Numerical simulation of U(N)×U(N) ABJM on S 3 Motivation： CFT 3 ( k: Chern-Simons level ) / Ad. S 4 　　relatively easy ＡＢＪＭ theory [Aharony-Bergman-Jafferis-Maldacena ’ 08] 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… Monte Carlo calculation of the Free energy in U(N)×U(N) ABJM theory on S 3（with keeping all symmetry） ・Test all known analytical results ・Relation between the known results and our simulation result 4

Developments on ＡＢＪＭ Free energy ・June 2008： ABJM was born. [Aharony-Bergman-Jafferis-Maldacena] ・July 2010： Planar limit for strong coupling [Drukker-Marino-Putrov] Agrees with SUGRA’s result!! [Cf. Cagnazzo-Sorokin-Wulff ’ 09] ※CP３ has nontrivial 2 -cycle ～string wrapped on CP 1⊂ CP 3＝worldsheet instanton ? ・November 2010: Calculation for k=fixed, N→∞ [Herzog-Klebanov-Pufu-Tesileanu] Formally same (※ λ=N/k)

（Cont’d）Development on ABJM free energy ・June 2011 : Summing up all genus around planar limit for strong λ [Fuji-Hirano-Moriyama] Formally same ・October 2011： Exact calculation for N=2 [Okuyama] ・October 2011： Calculation for k<<1, k<<N where [Marino-Putrov] 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, [ Hanada-M. H. -Honma-Nishimura-Shiba-Yoshida] Free energy is a smooth function of k !!

Contents 1. Introduction & Motivation 2. How to put ABJM on a computer 3. Result 4. Interpretation 5. Summary & Outlook 7

How do we ABJM on a computer? ～Approach by the orthodox method（=Lattice）～ Action： Difficulties in “formulation” ・It is not easy to construct CS term on a lattice [Cf. Bietenholz-Nishimura ’ 00] ・It is generally difficult to treat SUSY on a lattice [Cf. Giedt ’ 09] Practical difficulties ・∃Many fermionic degrees of freedom → Heavy computational costs ・CS term = purely imaginary → sign problem hopeless… 8

(Cont’d)How do we put ABJM on a computer? Lattice approach is hopeless… We can apply the localization method for the ABJM partition function 9

Localization method Original partition function: [Cf. Pestun ’ 08] where 1 parameter deformation: Consider t-derivative: Assuming Q is unbroken We can use saddle point method!!

(Cont’d) Localization method Consider fluctuation around saddle points: where 11

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

(Cont’d) Localization of ABJM theory Saddle point: Gauge 1 -loop Matter 1 -loop CS term

(Cont’d)How do we put ABJM on a computer? After applying the localization method, the partition function becomes just 2 N-dimentional integration: Sign problem Further simplification occurs!! 14

Simplification of ABJM matrix model [Kapustin-Willett-Yaakov ’ 10, Okuyama ‘ 11, Marino-Putrov ‘ 11] Cauchy identity: Fourier trans. : 15

(Cont’d) Simplification of ABJM matrix model Gaussian integration Fourier trans. : Cauchy id. : 16

Short summary Lattice approach is hopeless… (∵SUSY, sign problem, etc) Localization method Complex Cauchy identity, Fourier trans. ≠probability & Gauss integration Easy to perform simulation even by our laptop 17

How to calculate the free energy Problem: Monte Carlo can calculate only expectation value We regard the partition function as an expectation value under another ensemble: Note: VEV under the action:

Contents 1. Introduction & Motivation 2. How to put ABJM on a computer 3. Result 4. Interpretation 5. Summary & Outlook 19

Warming up: Free energy for N=2 There is the exact result for N=2: [ Okuyama ’ 11] for odd k for even k F (free energy) Complete agreement with the exact result !! k ( CS level )

Result for Planar limit [Drukker-Marino-Putrov ’ 10] ・Weak couling： ・Strong coupling： Worldsheet instanton Weak coupling Different from worldsheet instanton behavior Strong coupling strong weak

3/2 power low in 11 d SUGRA limit [Drukker-Marino-Putrov ‘ 10, Herzog-Klebanov-Pufu-Tesileanu ‘ 10] 11 d classical SUGRA: F/N 3/2 F N 3/2 1/N

(Cont’d) 3/2 power low in 11 d SUGRA limit 11 d classical SUGRA: Perfect agreement !!

Comparison with Fuji-Hirano-Moriyama [Fuji-Hirano-Moriyama ’ 11] Ex. ) For N=4 Weak coupling FHM Discrepancy Ｎ andcoupling dependent on ｋ Almost agreesindependent with FHM forofstrong →different from instanton (～exp dumped) →more precise comparison by taking difference strong weak bahavior Almost agrees with FHM for strong coupling →more precise comparison by taking difference

Contents 1. Introduction & Motivation 2. How to put ABJM on a computer 3. Result 4. Interpretation 5. Summary & Outlook 25

Fermi gas approach [Marino-Putrov ’ 11] Cauchy id. : Result: Regard as a Fermi gas system where Our result says that this remains even for large k? ?

Origin of Discrepancy for the Planar limit (without MC) [Marino-Putrov ‘ 10] Analytic continuation: [Cf. Yost ’ 91, Dijkgraaf-Vafa ‘ 03] Lens space L(2, 1)=S 3/Z 2 matrix model: Genus expansion: 27

(Cont’d))Origin of Discrepancy for the Planar limit (without MC) The “derivative” of planar free energy is exactly found as We impose the boundary condition: By using asymptotic behavior, Necessary for satisfying b. c. , taken as 0 for previous works Cf. [Drukker-Marino-Putrov ‘ 10]

Origin of discrepancy for all genus Discrepancy is fitted by This is explained by ``constant map’’ contribution in language of topological string: [ Bershadsky-Cecotti-Ooguri-Vafa ’ 93, Faber-Pandharipande ’ 98, Marino-Pasquwtti-Putrov ’ 09 ] Divergent, but Borel summable: 29

Comparison with discrepancy and Fermi gas Divergent, but Borel summable: genus 2 Borel sum of Constant map realizes Fermi Gas（small k）result！！ →Can we understand the relation analytically?

Fermi Gas from Constant map contribution： Borel Expand around k=0 True for all k? All order form？ 　 Agrees with Fermi Gas result！ →Fermi Gas result is asymptotic series around k=0 31

Contents 1. Introduction & Motivation 2. How to put ABJM on a computer 3. Result 4. Interpretation 5. Summary & Outlook 32

Summary Monte Carlo calculation of the Free energy in U(N)×U(N) ABJM theory on S 3（with keeping all symmetry） ・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. ・The free energy for whole region up to instanton effect： where ・Predict all order form of Fermi Gas result： ～instanton effect

Problem ・What is a physical meaning of constant map contribution? - In Fermi gas description, this is total energy of membrane instanton [ Becker-Strominger ‘ 95] - Why is ABJM related to the topological string theory? ・If there is also constant map contribution on the gravity side, there are α’-corrections at every order of genus - Does it contradict with the proof for non-α’-correction? [ Kallosh-Rajaraman ’ 98] - Is constant map origin of free energy on the gravity side? ? ・Mismatch between renormalization of ‘t Hooft coupling and Ad. S radius [ Bergmanr-Hirano ’ 09] 34

Outlook Monte Carlo method is very useful to analyze unsolved matrix models. In particular, there are many interesting problems for matrix models obtained by the localization method. Example(3 d): ・Other observables Ex. ) BPS Wilson loop ・Other gauge group ・On other manifolds Ex. ) Lens space ・Other theory Ex. ) ABJ theory ・Nontrivial test of 3 d duality ・Nontrivial test of F-theorem for finite N [ Hanada-M. H. -Honma-Nishimura-Shiba-Yoshida, work in progress] [ M. H. -Imamura-Yokoyama, work in progress] [ Azeyanagi-Hanada-M. H. -Shiba, work in progress] [ M. H. -Honma-Yoshida, work in progress] Example(4 d): ・ Example(5 d): ・ 35

Appendix 37

“Direct” Monte Carlo method(≠Ours). . . . . ① Distribute random numbers many times . . Ex. ) The area of the circle with the radius 1/2 . . ② Count the number of points which satisfy ③ Estimate the ratio Note: This method is available only for integral over compact region 38

“Markov chain” Monte Carlo (=Ours) Ex. ) Gaussian ensemble (by heat bath algorithm) ① Generate random configurations with Gaussian weight many times We can generate the following Markov chain from the uniform random numbers: ② Measure observable and take its average 39

Essence of Markov chain Monte Carlo Consider the following Markov process: “sweep” Under some conditions, transition prob. monotonically converges to an equilibrium prob. “thermalization” We need an algorithm which generates “Hybrid Monte Carlo algorithm” is useful !! 40

Hybrid Monte Carlo algorithm (Detail is omitted. Please refer to appendix later. ) ① Take an initial condition freely [ Duane-Kennedy-Pendleton-Roweth ’ 87] [ Cf. Rothe, Aoki’s textbook] Regard as the “conjugate momentum” ② Generate the momentum with Gaussian weight ③ Solve “Molecular dynamics” “Hamiltonian”: ④ Metropolis test accepted with prob. rejected with prob.

Note on Statistical Error Average: If all configurations were independent of each other, 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. ) 42

Taking planar limit

N=8 44

Higher genus 45