1. Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant
Masazumi Honda
SOKENDAI & KEK
Reference: JHEP (2012)
(arXiv: [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

2. F N3/2 Introduction we can realize this result even by our laptop
(free energy)
N3/2
Surprisingly,
we can realize this result even by our laptop

3. CFT3 AdS4 / ＡＢＪＭ theory Numerical simulation of U(N)×U(N) ABJM on S3
Motivation：
( k: Chern-Simons level )
CFT3
AdS4 /
　　relatively easy
ＡＢＪＭ theory
（Intermediate）
Extremely difficult！
Key for a relation
between string and M-theory?
[Aharony-Bergman-Jafferis-Maldacena ’08]
vestigatio
relatively hard
Investigate the whole region by numerical simulation!

4. This talk is about… 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

5. 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 CP1⊂ CP3＝worldsheet instanton ?
・November 2010: Calculation for k=fixed, N→∞
[Herzog-Klebanov-Pufu-Tesileanu]
Formally same
(※ λ=N/k)

6. （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
[Marino-Putrov]
where
Correction to Airy function
→How about for large k??
・February 2012： Numerical simulation in the whole region(=this talk）
[ Hanada-M.H.-Honma-Nishimura-Shiba-Yoshida]
At least up to instanton effect, for all k,
Free energy is a smooth function of k !!

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

8. 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
・It is generally difficult to treat SUSY on a lattice
[Cf. Bietenholz-Nishimura ’00]
[Cf. Giedt ’09]
Practical difficulties
・∃Many fermionic degrees of freedom → Heavy computational costs
・CS term = purely imaginary → sign problem
hopeless…

9. (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

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

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

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

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

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

15. Simplification of ABJM matrix model
[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.:

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

18. 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:
VEV under the action:
Note:

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

20. 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 )

21. 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
strong
weak

22. 3/2 power low in 11d SUGRA limit
[Drukker-Marino-Putrov ‘10, Herzog-Klebanov-Pufu-Tesileanu ‘10]
11d classical SUGRA:
F/N3/2 F
N3/2
1/N

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

24. Comparison with Fuji-Hirano-Moriyama
Ex.) For N=4
Weak coupling
FHM
Discrepancy independent of Ｎ and dependent on ｋ
→different from instanton bahavior (～exp dumped)
Almost agrees with FHM for strong coupling
→more precise comparison by taking difference
Almost agrees with FHM for strong coupling
→more precise comparison by taking difference
strong
weak

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

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

27. 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)=S3/Z2 matrix model:
Genus expansion:

28. (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
We impose the boundary condition:
Cf.
By using asymptotic behavior,
Necessary for satisfying b.c. ,
taken as 0 for previous works

29. 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:

30. 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

31. Fermi Gas from Constant map
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

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

33. Summary Monte Carlo calculation of the Free energy
in U(N)×U(N) ABJM theory on S3（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
～instanton effect
・Predict all order form of Fermi Gas result：

34. Problem ・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?
・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 AdS radius
[ Bergmanr-Hirano ’09]

35. Outlook ・Other observables Ex.) BPS Wilson loop ・Other gauge group
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(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]
[ M.H.-Imamura-Yokoyama, work in progress]
[ Azeyanagi-Hanada-M.H.-Shiba, work in progress]
[ M.H.-Honma-Yoshida, work in progress]
Example(4d): ・
Example(5d): ・

36. 完

37. Appendix

38. “Direct” Monte Carlo method(≠Ours)
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

39. “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

40. 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 !!

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

42. 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. )

43. Taking planar limit

44. N=8

45. Higher genus