8/1(Thu), 2013 Parallel talk (Theoretical development) Daisuke Kadoh (KEK) D.K. and Syo Kamata, in preparation TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA Lattice simulation of lower dimensional SYM with 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 duality from numerical simulation using lattice gauge theory Target: 1-dim SYM with 16 supercharges :

S. Catterall and T. Wiseman JHEP 0712 (2007) 104 Phys. Rev. D 78, (2008) … J. Nishimura, M. Hanada et al. Lattice Non-lattice non-compact gauge field (with gauge fixing) & momentum sharp cutoff PRL 99 (07) PRL 100 (08) …… 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 calculated from gravity side, t 1996 Klebanov -Tseytlin 1998 Itzhaki-Maldacena -Sonnenschein-Yankielowicz N D0-branes We derive the coefficient from gauge theory side by 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 operator given 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.