1. Non-lattice simulation of supersymmetric gauge theories as a probe to quantum black holes and strings
Jun Nishimura (KEK)
Plenary talk at LATTICE 2009, Beijing, China, July 30, ’09.
Ref.) Hanada-J.N.-Takeuchi, PRL 99 (’07)
Anagnostopoulos-Hanada- J.N.-Takeuchi, PRL 100 (’08)
Hanada-Miwa-J.N.-Takeuchi, PRL 102 (’09)
Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09)
Ishiki-Kim-J.N.-Tsuchiya, PRL 102 (’09)

2. 0. Introduction

3. Gauge/gravity duality (a conjecture from superstring theory)
Maldacena (’97)
6225 cites on SPIRES
U(N) supersymmetric Yang-Mills theory (SYM)
with 16 (32) supercharges
a classical solution of 10d supergravity (SUGRA)
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

4. 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
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

5. Jun Nishimura (KEK) Non-lattice simulation ...
Why interesting ?
realization of an old idea by ’t Hooft (’73)
emergent space-time
applications:
ultimate goal:
strong-weak duality
large N gauge theory string theory
anti-de Sitter space,
black hole geometry at finite T
hadron and condensed matter physics, from curved space-time
black hole thermodynamics, from gauge theory
non-perturbative & background indep. formulation of string theory
Studies of strongly coupled gauge theory are important !
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

6. Jun Nishimura (KEK) Non-lattice simulation ...
Monte Carlo studies
supersymmetry
translational symmetry
broken by the lattice
SUSY can be restored in the continuum limit.
(Typically, fine-tuning is necessary.)
recent developments in “lattice SUSY”
lattice actions with various symmetries
e.g.) preserving 1 supercharge using topological twist
non-lattice simulations of SYM with 16 supercharges
1d gauge theory
extension to 3d, 4d gauge theory
using the idea of large N reduction
Catterall’ talk, Rm.2, 14:30-
superconformal
(AdS/CFT) 32
c.f.) Kaplan-Unsal (’05), Elliott-Giedt-Moore (’08)
Catterall (’08)
No fine-tuning !
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

7. Jun Nishimura (KEK) Non-lattice simulation ...
Plan of the talk
0. Introduction
1. Non-lattice simulation of 1d SYM with 16 supercharges
black hole thermodynamics from 1d SYM
Schwarzschild radius from Wilson loop
2. Extension to higher dim. (based on large-N reduction)
　　　　　preliminary results for Wilson loop and 2pt correlators
3. Summary
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

8. 1. Non-lattice simulation of 1d SYM with 16 supercharges

9. Jun Nishimura (KEK) Non-lattice simulation ...
1d SYM with 16 supercharges
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)
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

10. Jun Nishimura (KEK) Non-lattice simulation ...
Hanada-J.N.-Takeuchi, PRL 99 (07) [arXiv: ]
Note: Gauge symmetry can be fixed non-perturbatively in 1d.
static diagonal gauge :
residual gauge symmetry :
should be fixed by imposing
RHMC algorithm can be used efficiently
(Fourier acceleration without extra cost etc.)
c.f.) lattice approach : Catterall-Wiseman, JHEP 0712:104,2007
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

11. Jun Nishimura (KEK) Non-lattice simulation ...
Result: 　Polyakov line
Anagnostopoulos-Hanada- J.N.-Takeuchi, PRL 100 (’08) [arXiv: ]
High T expansion
(including next-leading order)
Characteristic behavior of
the deconfined phase
no phase transition unlike in bosonic case
consistent with analyses on the gravity size (Barbon et al., Aharony et al.)
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

12. Jun Nishimura (KEK) Non-lattice simulation ...
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
quantum description of the states inside the BH
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

13. Result: Internal energy
Anagnostopoulos-Hanada- J.N.-Takeuchi, PRL 100 (’08) [arXiv: ]
free energy
high T expansion
(incl. next-leading order)
result obtained
from 10d BH
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

14. More precise comparison
Hanada-Hyakutake-J.N.-Takeuchi,
PRL 102 (’09) [arXiv: ]
corrections
slope = 4.6
finite cutoff effects
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

15. 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
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

16. Jun Nishimura (KEK) Non-lattice simulation ...
Result: Wilson loop
Hanada-Miwa-J.N.-Takeuchi, PRL 102 (’09) [arXiv: ]
Schwarzschild radius
from the Wilson loop
high T exp.
(next-leading)
subleading term
(perturbative corrections)
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

17. Prediction from gauge/gravity duality (III)
predicted from dual geometry by Sekino-Yoneya (’99)
using Gubser-Klebanov-Polyakov-Witten relation (’98)
power law behavior
reproduced from 1d SYM
work in progress
Hanada-J.N.-Sekino-Yoneya
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

18. 2. Extension to higher dimensions based on large-N reduction

19. Jun Nishimura (KEK) Non-lattice simulation ...
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)
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

20. A novel large N reduction for 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
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

21. A novel large N reduction for SYM on (cont’d)
Ishiki-Ishii-Shimasaki
-Tsuchiya (’08)
SYM on
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

22. A check in the weak coupling limit
Ishiki-Kim-J.N.-Tsuchicya
Phys.Rev.Lett.102 (09) [arXiv: ]
deconfinement transition at finite T
known results
for SYM on
(Aharony et al. ’03)
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

23. circular Wilson loop in
conformal
mapping
grand circle
Erickson-Semenoff-Zarembo (’00),
Drukker-Gross (’00),
Pestun (’07)
(agree with the result from dual geometry)
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

24. Result: circular Wilson loop
(preliminary) 1 2
work in progress
Honda-Ishiki-J.N.-Tsuchiya
strong coupling
weak coupling
all order
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

25. Jun Nishimura (KEK) Non-lattice simulation ...
Chiral primary operators
e.g.)
2-pt function
large-N reduction
SUSY non-renormalization theorem
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

26. Jun Nishimura (KEK) Non-lattice simulation ...
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.
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

27. 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 to conformal symmetry
i.e.,
Gauge/gravity duality predicts :
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...

28. 3. Summary

29. Jun Nishimura (KEK) Non-lattice simulation ...
Summary
non-lattice simulations of
supersymmetric large N gauge theories
revealed direct connections to gravity as predicted by gauge/gravity duality
models with maximal supersymmetries
16 supercharges (1d SYM)
32 supercharges
( SYM on )
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)
LATTICE 2009
Jun Nishimura (KEK) Non-lattice simulation ...