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.
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