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)
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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