1. Gauge/gravity duality ・ Strongly coupled gauge theory ( in large N limit ) ・ String theory gravity in a curved space AdS/CFT [1997, Maldacena] D-brane graviton ： closed string ： open string gauge field

2. Applications of gauge/gravity duality 1. Non-perturbative formulation of string theory? 2. Quantum behavior of black holes “resolving black hole information loss problem” 3. Holographic QCD, Holographic superconductor, … “understanding of strongly coupled theories from gravity side” The duality is not a theorem, but a conjecture. Verification is important.

3. New frontier in lattice gauge theories Gauge/gravity duality [1997, Maldacena,…] New studies which bridge the gap between gauge theories and the theories of gravity. Lattice supersymmetry [2004, Kaplan, Sugino,…]

4. Gauge/gravity duality － From lattice gauge theory to Black hole Sakata Memorial KMI Workshop on SCGT15 ＠ Sakata-Hirata Hall, Nagoya Univ., March 5, 2015 Daisuke Kadoh (KEK) D.K. and S.Kamata, PoS LATTICE 2012:064, PoS(LATTICE 2013) 343 D.K. and S.Kamata, in preparation

5. Talk plan １. Motivation ２. １ -dim SYM and gauge/gravity duality ３. 1-dim SYM on the lattice - Lattice action - SUSY Ward-Takahashi identity ４． Our lattice result of internal energy of black hole ５． Summary

6. ２．１ -dim SYM with 16 supercharges and gauge/gravity duality

7. Continuum action Sixteen SUSY-invariance 1-dim SYM with 16 supercharges : sixteen Grassmann parameters : scalar fields : gauge field : fermi fields

8. 1D SYM with 16 supercharges Thermodynamics of black hole Gauge/gravity duality in 1-dim SYM N D0-branes solution in IIA superstring t N D0-branes black hole 1996 Klebanov -Tseytlin BH internal energy in gravity side large We compute physical quantities of black hole from lattice simulations of gauge theory side to test the duality.

9. Previous numerical results (1) Non-lattice M.Hanada, J. Nishimura et al. (2007- 2009) temperature Good agreement with gravity side Black hole internal energy Regularized by momentum sharp cutoff

10. Previous numerical results (2) Lattice S. Catterall and T. Wiseman (2007-2010) Compact link field (exact gauge invariance) SUSYs are broken by lattice cut-off Disagreement …? Lattice is inferior to the simple momentum sharp cutoff ?

11. Symmetries at regularized level 2-chargesSUSY gauge sym. (BRST sym.) Non-lattice (Nishimura et al) Lattice (Catterall et al) NoneOK Lattice (this research) We employ a lattice action with two exact supercharges, proposed by F.Sugino [Sugino, 2005]. None

12. ２． 1-dim SYM on the lattice

13. Q-exact action Lattice action up to gauge transformation Two exact supercharges [2005, F.Sugino] discretization two supercharges

14. SUSY WTI Numerical results of SUSY Ward-Takahashi identity SUSY WTI holds.All sixteen supercharges are recovered for a<<1.

15. ４． Internal energy of black hole

16. Lattice result of internal energy of black hole Kawahara-Takeuchi-Nishimura(2007), D.K. and Kamata (in preparation) HTE (NLO,N=14) prediction in gravity side (1996 Klebanov -Tseytlin)

17. Low temperature -Fit formula at NLO [Nishimura’s non-lattice result(2009)] [gravity side] [our result]

18. Polyakov loop, SU(3) Reconsideration of Catterall-Wiseman’s results “High temperature” “Low temperature” Catterall-WisemanThis study

19. ５． Summary

20. Summary 1d SYM with 16 supercharges N D0-branes in type IIA superstring Sugino’s lattice action with two exact supercharges. Numerical results of SUSY WTI suggested us that the lattice action reproduces correct continuum theory. Internal energy of black hole

21. Future directions Simulation of 1d SYM with 16 supercharges  Continuum limit of NLO coefficient  Explanation of leading order coefficient, 7.41  Other physical quantities(Wilson loop,…)  1/N corrections Verification of gauge/gravity duality in higher dimensions 2-dim SYM N D1-branes in type IIB superstring Lattice simulations, D.K. and Eric Giguere

22. NLO Nishimura’s non-lattice result(2009) : Our result :