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Numerical tests of the gauge-gravity duality and beyond

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### Numerical tests of the gauge-gravity duality and beyond

### 2. Monte Carlo simulation of 1d SYM with 16 supercharges

### 3. Black hole thermodynamics from 1d SYM

### corrections and string loop corrections

### 5. Wilson loop and correlation functions

Jun Nishimura (KEK, SOKENDAI)

KEK, lattice meeting for future prospects, KEK, Feb. 15, 2010.

Ref.) Hanada-J.N.-Takeuchi, PRL 99 (’07) 161602

Anagnostopoulos-Hanada- J.N.-Takeuchi, PRL 100 (’08) 021601

Hanada-Miwa-J.N.-Takeuchi, PRL 102 (’09) 181602

Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602

Ishiki-Kim-J.N.-Tsuchiya, PRL 102 (’09) 111601

Hanada-J.N.-Sekino-Yoneya, arXiv:0911.1623[hep-th]

Increasingly many applications of the gauge/gravity duality

strongly coupled gauge theory classical gravity theory

quantum

- hadron physics

confined phase

(Sakai-Sugimoto model etc.)

mass spectrum, effective coupling, nuclear forces,…

deconfined phase (quark-gluon plasma)

transport coefficients (viscosity etc.),…

- condensed matter physics

superfluidity, superconductivity, quantum Hall systems,…

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

But, to what extent does the duality really hold quantitatively ?

- AdS/CFT

4d SU(N) SYM theory with maximal SUSY

(super) Conformal Field Theory

- large N lim.
- strong coupling lim.

string loop corrections

can be neglected

Classical gravity theory becomes valid.

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Monte Carlo simulation of 1d SYM with 16 supercharges

precision test of the gauge/gravity duality from first principles (non-conformal case)

- internal energy v.s. temperature

microscopic origin of black hole thermodynamics

- Wilson loop

One can see Schwarzschild radius from gauge theory!

- correlation functions

confirmation of the GKP-Witten prescription

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Plan of the talk

- 0. Introduction
- 1. An intuitive argument for the gauge/gravity duality
- 2. Monte Carlo simulation of 1d SYM with 16 supercharges
- 3. Black hole thermodynamics from 1d SYM
- 4. corrections and string loopcorrections
- 5. Wilson loop and correlation functions
- 6. Extension to higher dimensions based on large-N reduction
- 7. Summary

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Gauge/gravity duality (a conjecture from superstring theory)

Maldacena (’97)

6607 cites on SPIRES

U(N) supersymmetric Yang-Mills theory (SYM)

with 16 (32) supercharges

a classical solution of 10d supergravity (SUGRA)

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

An intuitive argument for the conjecture

D brane “soliton solution”

in superstring theory

(Dirichlet)

emission of a graviton

propagation of a gauge particle

dim U(N) SYM

low energy limit

curved 10d space-time

N

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Hanada-J.N.-Takeuchi, PRL 99 (’07) 161602

1d gauge theory

p.b.c.

anti p.b.c.

(without loss of generality)

low T

strongly coupled

dual gravity description

high T

weakly coupled high T exp.

Kawahara-J.N.-Takeuchi (’07)

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Non-lattice simulation

Hanada-J.N.-Takeuchi, PRL 99 (07) 161602 [arXiv:0706.1647]

Note: Gauge symmetry can be fixed non-perturbativelyin 1d.

- static diagonal gauge :

- residual gauge symmetry :

RHMC algorithm can be used efficiently

(Fourier acceleration without extra cost etc.)

c.f.) lattice approach : Catterall-Wiseman, JHEP 0712:104,2007

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Anagnostopoulos-Hanada- J.N.-Takeuchi, PRL 100 (’08) 021601

quantum description of the states inside the BH

Prediction from gauge/gravity duality (I)

- dual geometry

Hawking’s theory

black hole thermodynamics

7.41

Klebanov-Tseytlin (’96)

Gauge/gravity duality predicts that

this should be reproduced by 1d SYM. large-N, low T

microscopic origin of the black hole thermodynamics

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

After taking the decoupling limit :

range of validity:

Black hole thermodynamics

Hawking temperature :

Bekenstein-Hawking entropy :

Klebanov-Tseytlin (’96)

7.41

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Result: Internal energy

Anagnostopoulos-Hanada- J.N.-Takeuchi, PRL 100 (’08) 021601 [arXiv:0707.4454]

free energy

high T expansion

(incl. next-leading order)

result obtained

from 10d BH

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Digression : 8 supercharge case

Hanada, Matsuura, J.N., Robles-Llana, in progress

- b=2 obtained from Born-Oppenheimer
- analysis in gauge theory by A.Smilga (’09)
- What is the gravity dual ?

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602

Ishiki-Hanada-Hyakutake-J.N., work in progress

corrections to SUGRA action

low energy effective action of type IIA superstring theory

tree-level scattering amplitudes of the massless modes

leading term : type IIA SUGRA action

explicit calculations of 2-pt and 3-pt amplitudes

4-pt amplitudes

Complete form is yet to be determined,

but we can still make a dimensional analysis.

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Black hole thermodynamics with corrections

curvature radius of the dual geometry

More careful treatment leads to the same conclusion.

(Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

confronting Monte Carlo data

Hanada-Hyakutake-J.N.-Takeuchi,

PRL 102 (’09) 191602 [arXiv:0811.3102]

corrections

slope = 4.6

finite cutoff effects

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

string loop corrections to SUGRA action

known from IIA string theory

1-loop

Bern, Rozowsky, Yan

Bern, Dixon, Dunbar, Perelstein, Rozowsky

2-loop

Using Kawai-Lewellen-Tye relation to SYM amp.

Only meta-stable vacuum

at finite N (flat directions)

negative specific heat !

small N, very low T regime

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

1/N corrections (preliminary results)

Ishiki-Hanada-Hyakutake-J.N., work in progress

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Hanada-Miwa-J.N.-Takeuchi, PRL 102 (’09) 181602

Hanada-J.N.-Sekino-Yoneya, arXiv:0911.1623[hep-th]

Prediction from gauge/gravity duality (II)

SYM

probe D-brane

Maldacena (’98)

Rey-Yee (’98)

string

Wilson loop

minimal surface

- Schwarzschild radius from Wilson loop (in 1d SYM)

1.89

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Hanada-Miwa-J.N.-Takeuchi, PRL 102 (’09) 181602 [arXiv:0811.2081]

Schwarzschild radius

from the Wilson loop

high T exp.

(next-leading)

subleading term

(perturbative corrections)

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Prediction from gauge/gravity duality (III)

correlation functions in gauge theory

generating functional

operator-field correspondence

gauge gravity

Gubser-Klebanov-Polyakov-Witten relation (’98)

SUGRA action evaluated at the classical solution

with the boundary condition

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Result: 2pt correlation functions

e.g.)

predicted from dual geometry by Sekino-Yoneya (’99)

using Gubser-Klebanov-Polyakov-Witten relation (’98)

power law behavior

reproduced from 1d SYM

Hanada-J.N.-Sekino-Yoneya,

arXiv:0911.1623[hep-th]

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Large N reduction

Eguchi-Kawai (’81)

Bhanot-Heller-Neuberger (’82)

Gonzalez-Arroyo & Okawa (’82)

U(N) gauge theory in D-dim. torus

reduce

to a point

large-N reduced model

c.f.) QCD at

Narayanan-Neuberger (’03)

Bringoltz-Sharpe (’09)

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

A novel large N reductionfor SYM on

Ishiki-Ishii-Shimasaki

-Tsuchiya (’08)

reduce

to a point

1d SYM + mass deformation

(preserving 16 SUSY)

many classical vacua preserving 16 SUSY

all degenerate

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

A novel large N reductionfor SYM on (cont’d)

Ishiki-Ishii-Shimasaki

-Tsuchiya (’08)

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

A check in the weak coupling limit

Ishiki-Kim-J.N.-Tsuchicya

Phys.Rev.Lett.102 (09) 111601[arXiv:0810.2884]

deconfinement transition at finite T

known results

for SYM

on

(Aharony et al. ’03)

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

circular Wilson loop in

conformal

mapping

grand circle

Erickson-Semenoff-Zarembo (’00),

Drukker-Gross (’00),

Pestun (’07)

(agree with the result from dual geometry)

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

2

Result: circular Wilson loop(preliminary)

work in progress

Honda-Ishiki-J.N.-Tsuchiya

strong coupling

weak coupling

all order

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Chiral primary operators

2-pt function

large-N reduction

SUSY non-renormalization theorem

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Result : 2pt function of chiral primary operators

(preliminary)

work in progress

Honda-Ishiki-Kim-J.N.-Tsuchiya

free theory result for the reduced model for N=14

result for SYM

indication of SUSY non-renorm. th.

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

More interesting quantities

- rectangular Wilson loop, higher point functions

in SYM on

c.f.) Berenstein-Cotta-Leonardi (’08)

calculable by using the large-N reduced model

highly non-trivial test of AdS/CFT

Due toconformal symmetry

i.e.,

Gauge/gravity duality predicts :

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Summary and future prospects

- Monte Carlo simulations of

1d U(N) SYM with 16 supercharges

Precision test of the gauge/gravity duality from first principles (non-conformal case)

negative specific heat ?

Black hole thermodynamics including corrections

Schwarzschild radius from Wilson loop

GKP-Witten prescription for correlation functions

- gauge/gravity duality seems to hold including

corrections

non-perturbative definition of superstring theory

on a particular curved background

- 1d U(N) SYM with 8 supercharge

E v.s. T relation shows clear power law (p=2); gravity dual ?

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

Summary and future prospects

- extension to models in higher dimensoins

1d SYM16 supercharges

SYM on

32 supercharges

Superconformal !

a novel large-N reduction

(preserving 16 SUSY)

restoration of 32 SUSY at large N : yet to be seen

at strong coupling

SYM on , theories with less SUSY (easier)

- IKKT matrix model (Ishibashi-Kawai-Kitazawa-Tsuchiya ’96)

0d reduction of the 1d SYM with 16 superchages

Emergence of 4d space-time from non-perturbative superstring theory

Anagnostopoulos, Aoyama, Azuma, Hanada, J.N.

Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

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