LATTICE2013@Mainz, 8/1(Thu), 2013

Parallel talk (Theoretical development)

Daisuke Kadoh (KEK) 　

D.K. and Syo Kamata, in preparation

Lattice simulation of lower dimensional SYMwith sixteen supercharges

1. Motivation

　 　 　 　 　

Recent progress of Lattice supersymmetry

Gauge/Gravity duality (AdS/CFT,...)

Holographic QCD, Holographic superconductor…

However, this is a conjecture.

Strongly coupled gauge theory in large N limit

Many applications:

Classical gravity on a curved space

Numerical verification of gauge/gravity dualityfrom numerical simulation using lattice gauge theory

Target: 1-dim SYM with 16 supercharges

:

S. Catterall and T. Wiseman

JHEP 0712 (2007) 104Phys. Rev. D 78, 041502(2008)…

J. Nishimura , M. Hanada et al.

Lattice

Non-lattice non-compact gauge field(with gauge fixing) & momentum sharp cutoff PRL 99 (07) 161602

PRL 100 (08) 021601……

Numerical simulations of 1-dim SYM with 16 supercharges

・ But, further simulations with sufficient statistics at low temperature are needed, because, for example, leading order temperature dependence of BH internal energy has not been explained from gauge side, yet.

・ Both group used regularized theories which break SUSY due to finite cutoff. In contrast, we employ a lattice theory with two exact supercharges.

1. Motivation

2. BH thermodynamics and N D0-branes solution

3. Continuum and lattice actions of 1-dim SYM with 16 supercharges 4. Results SUSY-WTI Black hole internal energy(Preliminary)

5. Summary

Talk’s Plan

2. BH thermodynamics and N D0-branes solution

Black hole: a solution of Einstein’ s eq.

String theory

Bekenstein-Hawking entropy

macroscopic picture

microscopic picture

1996 Strominger-Vafa

A: area of horizon

BH thermodynamics

・ Based on the gauge/gravity duality, the thermodynamics of black hole can be understood from gauge theory side.

N D0-branes solution

N D0-branes in type IIA superstring

1-dim SYM with 16 supercharges

BH internal energy can be analytically calculatedfrom gravity side,

t

1996 Klebanov -Tseytlin

1998 Itzhaki-Maldacena -Sonnenschein-Yankielowicz

N D0-branes

We derive the coefficient from gauge theory sideby lattice simulations to check conjectured duality.

3. 1-dim SYM with 16 supercharges

: nine scalars

・ Continuum action is given by

・ The action is also rewritten as Q-exact form,

‘t Hooft coupling

: gauge field

: sixteen fermions

up to gauge transformations.

Lattice action

covariant forward difference operatorgiven by the link field

・ Replacing and with and , respectively

・ We can define -transformations which satisfy

even on the lattice.

・ This is one dimensional version of lattice actions given by F.Sugino[2005].

up to gauge transformations,

Simulation details

・ We used HMC method to generate configuration,: , where is an auxiliary field to remove a cut-off order 4-fermi interaction;

・ The theory has a sign problem because pfaffian is generally complex. We treated the absolute value and the phase of pfaffian, individually.

・ The absolute value of pfaffian is given by the integral of pseudo fermion and rational approximation,

・ For the phase of pfaffian, we used phase quench, or phase reweighting.

SUSY Ward-Takahashi identity

c.f. 2007, Kanamori-Suzuki, for 2-dim N=(2,2) SYM

・ There are two SUSY breaking sources:

(1) 　 Finite temperature effect 　　　　　　　

(2) 　 Cut-off effect

physics

artifact

・ We have to show that the cut-off effect vanishes in the continuum limit under the existence of physical SUSY breaking source (temperature).

We examine “SUSY Ward Takahashi identity” numerically.

・ Adding SUSY breaking mass term to the action, the following partially breaking SUSY WTI holds at finite temperature,

for arbitrary operator O.: Supercurrent

Breaking term from the mass term

This figure shows that for .

Numerical results of SUSY WTI

・ We plotted a ratio, , which is given by lhs/rhs of the SUSY WTI,

.

and

Continuum and massless limit of SUSY WTI

・ We took the continuum limit of plateau’s values by performing constant fit of them with three different lattice spacings.

- massless limit

- continuum limit

・ Left figure shows that

and SUSY breaking cut-off effect vanishes in the continuum limit.

・ Taking the massless limit,

BH Internal Energy (Preliminary)

Kawahara-Nishimura-Takeuchi

Gravity side (1996 Klebanov -Tseytlin) NLO HTE at large Nc limit

・ Low temperature region of BH internal energy

- We performed NLO fit by the following formula,

[Non-lattice result by J.Nishimura et al. (2009)]

[Our result (preliminary) ]

5. Summary・ We aim to verify the gauge/gravity duality for 1-dim SYM with 16 supercharges( D0-branes of type IIA) by using lattice simulations.

・ Our lattice action holds two exact supercharges. Numerical result of SUSY WTI shows that SUSY breaking cut-off effect vanishes in the continuum limit. ・ Preliminary result of BH Internal energy is

・ It is also important to examine conjectured duality for other interesting quantities, for example, Polyakov line, Wilson line, etc.

As a next step, we have to take the continuum limit.

Thank you.