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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
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IntroductionF
(free energy)
N3/2Surprisingly,
we can realize this result even by our laptop
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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!
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This talk is about…
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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
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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
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( 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]
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Contents
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1. Introduction & Motivation2. How to put ABJM on a computer3. Result4. Interpretation5. Summary & Outlook
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How do we ABJM on a computer?
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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…
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(Cont’d)How do we put ABJM on a computer?
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We can apply the localization method for the ABJM partition function
Lattice approach is hopeless…
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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!!
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(Cont’d) Localization method
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Consider fluctuation around saddle points:
where
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Localization of ABJM theory[Kapustin-Willet-Yaakov ’09]
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(Cont’d) Localization of ABJM theory
Saddle point:
Matter 1-loop
Gauge 1-loop
CS term
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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!!
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Simplification of ABJM matrix model
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[Kapustin-Willett-Yaakov ’10, Okuyama ‘11, Marino-Putrov ‘11]
Cauchy identity:
Fourier trans.:
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(Cont’d) Simplification of ABJM matrix model
Gaussian integration
Fourier trans.:
Cauchy id.:
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Short summary
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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
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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:
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Contents
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1. Introduction & Motivation2. How to put ABJM on a computer3. Result4. Interpretation5. Summary & Outlook
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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 !!
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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
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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]
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(Cont’d) 3/2 power low in 11d SUGRA limit
11d classical SUGRA:
Perfect agreement !!
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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]
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Contents
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1. Introduction & Motivation2. How to put ABJM on a computer3. Result4. Interpretation5. Summary & Outlook
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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:
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Origin of Discrepancy for the Planar limit (without MC)
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[Marino-Putrov ‘10]
Analytic continuation:
Lens space L(2,1)=S3/Z2 matrix model:[Cf. Yost ’91, Dijkgraaf-Vafa ‘03]
Genus expansion:
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(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
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Origin of discrepancy for all genus
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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 ]
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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?
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Fermi Gas from Constant map
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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
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Contents
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1. Introduction & Motivation2. How to put ABJM on a computer3. Result4. Interpretation5. Summary & Outlook
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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 )
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Problem
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・ 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??
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Outlook
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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]
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完36
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Appendix
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“Direct” Monte Carlo method(≠Ours)
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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
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“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:
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Essence of Markov chain Monte Carlo
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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 !!
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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]
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Note on Statistical Error
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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. )
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Taking planar limit
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N=8
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Higher genus
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