matrix model of chern-simons matter theories beyond the spherical …seminar/pdf_2016_kouki/... ·...
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Shuichi Yokoyama
Yukawa Institute for Theoretical Physics
10. Jan. 2017 @ Osaka U
Matrix model of Chern-Simons matter theoriesbeyond the spherical limit
Ref. arXiv:1610.06471SY
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Infinitely many interacting CFT (conformal zoo).
④ (M-theory)
Interaction between membranes.
⑤ (3d CFT)
⑥ (AdS/CFT correspondence)
Dual CFT3 of (HS) gravity on AdS4
② (Mathematics)
Knot theory, Jones polynomial
Pure (HS) gravity on AdS3
[Witten ’89]
[BLG ’07, ABJM ’08]
③ (String theory)
Cubic string field theory, Open topological string theory
[Witten ’85]
Chern-Simons matter theory
[Moore_Seiberg ’89]
[Gaberdiel_Gopakumar ’11]
[Witten ’89]
① (Condensed matter physics)
Fractional quantum hall effect, anyon
[Klebanov-Polyakov ’02]
[Gopakumar-Vafa ’98]
[Zhan-Hansson-Kivelson ’88] [Fradkin-Lopez ’91]
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(i) S3
Partition function of CSM
(ii) S2 x S1
[Kapustin-Willet-Yaakov ’09] [Jafferis ’10] [Hama-Hosomichi-Lee ’10]
① ∃SUSY ⇒ Localization
[Bhattacharya-Minwalla ’08] [Kim ’09] [Imamura-Yokoyama ’09]
(superconformal index)
Exact
Exact
Path integral ⇒ Matrix model [Pestun ’07]
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(i) S3
Partition function of CSM
(ii) S2 x S1
[Kapustin-Willet-Yaakov ’09] [Jafferis ’10] [Hama-Hosomichi-Lee ’10]
① ∃SUSY ⇒ Localization
[Bhattacharya-Minwalla ’08] [Kim ’09] [Imamura-Yokoyama ’09]
(i) S3
(ii) S2 x S1
[Klebanov-Pufu-Safdi ’11]
② Non-SUSY ⇒ Large N limit
[Jain-Minwalla-Sharma-Takimi-Wadia-SY ’11]
(near critical high temperature)
(superconformal index)
Exact
Exact
Exact
Exact analysis is difficult for the S3 partition function...
Path integral ⇒ Matrix model [Pestun ’07]
perturbation
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(i) S3
Partition function of CSM
(ii) S2 x S1
[Kapustin-Willet-Yaakov ’09] [Jafferis ’10] [Hama-Hosomichi-Lee ’10]
① ∃SUSY ⇒ Localization
[Bhattacharya-Minwalla ’08] [Kim ’09] [Imamura-Yokoyama ’09]
(i) S3
(ii) S2 x S1
[Klebanov-Pufu-Safdi ’11]
② Non-SUSY ⇒ Large N limit
[Jain-Minwalla-Sharma-Takimi-Wadia-SY ’11]
(near critical high temperature)
(superconformal index)
Exact
Exact
Exact
Exact analysis is difficult for the S3 partition function...
Path integral ⇒ Matrix model [Pestun ’07]
⇒ Study a class of matrix model!
perturbation
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1. Introduction
Plan
6. Summary
2. The class of matrix models
3. Loop equation
✓
4. Solving the loop equation
4.1 planar solution
4.2 hole correction
5. Application to pure CS theory
4.3 genus one correction
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Matrix model of U(N)k CSM
① Consider partition function of CSM on S3[SY '16]
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Matrix model of U(N)k CSM
① Consider partition function of CSM on S3
② Expand the gauge field by vector spherical harmonics on S3
Lorenz gauge
Residual gauge except its Cartan part ⇒ σ
[SY '16]
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Matrix model of U(N)k CSM
① Consider partition function of CSM on S3
② Expand the gauge field by vector spherical harmonics on S3
Lorenz gauge
Residual gauge except its Cartan part ⇒ σ
③ Integrate out all the massive modes.
Expand the matter fields similarly.
[SY '16]
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Matrix model of U(N)k CSM
④ Pure Chern-Simons case ⇒ SUSY localization, Cohomological localization
[SY '16]
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Matrix model of U(N)k CSM
④ Pure Chern-Simons case ⇒ SUSY localization, Cohomological localization
⑤ Inclusion of the matter fields will be
[SY '16]
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(i) N=2 Chern-Simons theory with Nf fundamental chiral fields
Matrix model of U(N)k CSM
④ Pure Chern-Simons case ⇒ SUSY localization, Cohomological localization
⑤ Inclusion of the matter fields will be
[SY '16]
(ii) N=2 Chern-Simons theory with Nf pairs of anti/fundamental chiral fields
Ex.
⑥ Restrict the potential to consist of single trace operators:
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Matrix model of U(N)k CSM
⑦ Change the integration variables such that
[SY '16]
Logarithmic cut!
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This can be written by using a positive definite hermitian matrix Φ as
Matrix model of U(N)k CSM
⑦ Change the integration variables such that
[SY '16]
The goal is to solve this class of matrix models in the 1/N expansion.
Logarithmic cut!
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1. Introduction
Plan
6. Summary
2. The class of matrix models
3. Loop equation
✓
4. Solving the loop equation
4.1 planar solution
4.2 hole correction
4.3 genus one correction
5. Application to pure CS theory
✓
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ResolventDefinition
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ResolventDefinition
Once the resolvent is determined, the free energy is determined as follows.
where
The density function and the resolvent are related by
・
・
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ResolventDefinition
Once the resolvent is determined, the free energy is determined as follows.
where
The density function and the resolvent are related by
・
・
The correlator is also computed as follows.
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Schwinger-Dyson equationExpectation value of a generic operator O[φ]
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Schwinger-Dyson equationExpectation value of a generic operator O[φ]
Suppose the (infinitesimal) transformation
Consider a one-to-one transformation on R+ denoted by φs → φs'
Expand the right-hand side in terms of a
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Loop equationLet us choose
Then the transformation φs → φs' is one-to-one on R+.
for
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Loop equationLet us choose
Then the transformation φs → φs' is one-to-one on R+.
for
1st + 2nd =
3rd =
4th = 0
➡
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1. Introduction
Plan
6. Summary
2. The class of matrix models
3. Loop equation
✓
4. Solving the loop equation
4.1 planar solution
4.2 hole correction
4.3 genus one correction
5. Application to pure CS theory
✓
✓
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Solving the loop equationThe matrix model potential is generically written as
Assume
[SY '16]
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Solving the loop equationThe matrix model potential is generically written as
Assume
The consistent 1/N expansion of the resolvent will be
[SY '16]
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Solving the loop equationThe matrix model potential is generically written as
Assume
The consistent 1/N expansion of the resolvent will be
The loop equation boils down to
where
[SY '16]
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Solving the planar loop equationClaim The planar loop equation contains the saddle point equation in the planar limit.
Compute the discontinuity between x - iε and x + iε
➡
where
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Planar resolventThe support of the density function consists of s distinct connected intervals
Each interval corresponds to a square root cut of the quadratic loop equation.
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Planar resolventThe support of the density function consists of s distinct connected intervals
➡
Each interval corresponds to a square root cut of the quadratic loop equation.
Define a trial function such that
Assume (or construct a solution) that there exists a limit approaching the infinity such that the resolvent behaves as
➡
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Planar resolventThe support of the density function consists of s distinct connected intervals
➡
Each interval corresponds to a square root cut of the quadratic loop equation.
Define a trial function such that
Assume (or construct a solution) that there exists a limit approaching the infinity such that the resolvent behaves as
➡
➡
➡
Assume further that the trial function is analytic except supp(ρ0)
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Planar resolventDetermination of the 2s endpoints of the cut
① The asymptotic behavior of the resolvent
② Stability against the tunneling of eigenvalues between different cuts
[David '90]
[Jurkiewicz '90]
= “Equality of the Lagrange multiplier computed from the resolvent”
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Hole correction
This contains the saddle point equation:
The genus half loop equation
where
General solution is not known yet …
[SY '16]
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Genus 1 correction
where
The genus 1 loop equation
A general solution [Ambjorn et.al. '92] [Akemann '96]
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1. Introduction
Plan
6. Summary
2. The class of matrix models
3. Loop equation
✓
4. Solving the loop equation
4.1 planar solution
4.2 hole correction
4.3 genus one correction
5. Application to pure CS theory
✓
✓
✓
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Application to pure CS theory
Matrix model potential:
[SY '16]
➡ One cut solution!
w
W0
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Application to pure CS theory [SY '16]
Support of the density function:
Planar resolvent:
1st =
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Application to pure CS theory [SY '16]
Support of the density function:
Planar resolvent:
1st =
Edge of the cut:
➡
➡cf. [Marino '04]
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Application to pure CS theory [SY '16]
Potential vs eigenvalue distribution Resolvent result vs Past result
Planar result
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Application to pure CS theory [SY '16]
Genus 1 correction
Genus 1 correction:
“Moments”:
Free energy:
This is in precise agreement with the past exact result!!
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1. Introduction
Plan
6. Summary
2. The class of matrix models
3. Loop equation
✓
4. Solving the loop equation
4.1 planar solution
4.2 hole correction
4.3 genus one correction
5. Application to pure CS theory
✓
✓
✓
✓
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・ We presented explicit solution at the spherical order for a general potential and at the genus 1 order in the case where the potential does not have any 1/N correction.
・ We have derived the loop equation for all orders in the 1/N expansion including the hole correction.
Summary
・ We have performed a general analysis on a class of matrix models describing CSM theory on three sphere incorporating the standard technique of 1/N expansion.
・ We have applied the formulation to pure CS theory and confirmed that the presented solution reproduces the exact result known in the past.
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・ 2d CFT (or QFT?) description?
・ Exact analysis on three sphere partition function and 3d non-SUSY duality?
Future works
・ Relation to 2d bosonization?
・ Another iterative approach (topological expansion)?
・ AdS/CFT correspondence? Higher-spin (Vasiliev) theory?
cf. [Fukuma-Kawai-Nakayama '90] [Milnov-Morosov '90] [Dijkgraaf-Verlinde-Verlinde '91]
[Enyald '04]
・ Generalization to 2-matrices? Application to ABJM?
cf. [Marino-Putrov '10]
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・ Exact analysis on three sphere partition function and 3d non-SUSY duality?
Future works
・ Relation to 2d bosonization?
・ Another iterative approach (topological expansion)?
・ AdS/CFT correspondence? Higher-spin (Vasiliev) theory?
cf. [Fukuma-Kawai-Nakayama '90] [Milnov-Morosov '90] [Dijkgraaf-Verlinde-Verlinde '91]
[Enyald '04]
・ Generalization to 2-matrices? Application to ABJM?
cf. [Marino-Putrov '10]
Thank you!!
・ 2d CFT (or QFT?) description?