1. Numerical tests of the gauge-gravity duality and beyond
Jun Nishimura (KEK, SOKENDAI)
KEK, lattice meeting for future prospects, KEK, Feb. 15, 2010.
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)
Hanada-J.N.-Sekino-Yoneya, arXiv: [hep-th]

2. 0. Introduction

3. Increasingly many applications of the gauge/gravity duality
strongly coupled gauge theory classical gravity theory
quantum
hadron physics
confined phase
(Sakai-Sugimoto model etc.)
mass spectrum, effective coupling, nuclear forces,…
deconfined phase (quark-gluon plasma)
transport coefficients (viscosity etc.),…
condensed matter physics
superfluidity, superconductivity, quantum Hall systems,…
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

4. Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...
All this is nice!
But, to what extent does the duality really hold quantitatively ?
AdS/CFT
4d SU(N) SYM theory with maximal SUSY
(super) Conformal Field Theory
large N lim.
strong coupling lim.
string loop corrections
can be neglected
Classical gravity theory becomes valid.
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

5. Monte Carlo simulation of 1d SYM with 16 supercharges
precision test of the gauge/gravity duality from first principles (non-conformal case)
internal energy v.s. temperature
microscopic origin of black hole thermodynamics
Wilson loop
One can see Schwarzschild radius from gauge theory!
correlation functions
confirmation of the GKP-Witten prescription
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

6. Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...
Plan of the talk
0. Introduction
1. An intuitive argument for the gauge/gravity duality
2. Monte Carlo simulation of 1d SYM with 16 supercharges
3. Black hole thermodynamics from 1d SYM
corrections and string loop corrections
5. Wilson loop and correlation functions
6. Extension to higher dimensions based on large-N reduction
7. Summary
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

7. An intuitive argument for the gauge/gravity duality

8. Gauge/gravity duality (a conjecture from superstring theory)
Maldacena (’97)
6607 cites on SPIRES
U(N) supersymmetric Yang-Mills theory (SYM)
with 16 (32) supercharges
a classical solution of 10d supergravity (SUGRA)
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

9. An intuitive 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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

10. 2. Monte Carlo simulation of 1d SYM with 16 supercharges
Hanada-J.N.-Takeuchi, PRL 99 (’07)

11. Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...
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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

12. Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...
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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

13. 3. Black hole thermodynamics from 1d SYM
Anagnostopoulos-Hanada- J.N.-Takeuchi, PRL 100 (’08)

14. Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...
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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

15. Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...
Dual gravity description
After taking the decoupling limit :
range of validity:
Black hole thermodynamics
Hawking temperature :
Bekenstein-Hawking entropy :
7.41
Klebanov-Tseytlin (’96)
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

16. 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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

17. Digression : 8 supercharge case
Hanada, Matsuura, J.N., Robles-Llana, in progress
b=2 obtained from Born-Oppenheimer
analysis in gauge theory by A.Smilga (’09)
What is the gravity dual ?
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

18. corrections and string loop corrections
Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09)
Ishiki-Hanada-Hyakutake-J.N., work in progress

19. corrections to SUGRA action
low energy effective action of type IIA superstring theory
tree-level scattering amplitudes of the massless modes
leading term : type IIA SUGRA action
explicit calculations of 2-pt and 3-pt amplitudes
4-pt amplitudes
Complete form is yet to be determined,
but we can still make a dimensional analysis.
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

20. Black hole thermodynamics with corrections
curvature radius of the dual geometry
More careful treatment leads to the same conclusion.
(Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09)
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

21. confronting Monte Carlo data
Hanada-Hyakutake-J.N.-Takeuchi,
PRL 102 (’09) [arXiv: ]
corrections
slope = 4.6
finite cutoff effects
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

22. string loop corrections to SUGRA action
known from IIA string theory
1-loop
Bern, Rozowsky, Yan
Bern, Dixon, Dunbar, Perelstein, Rozowsky
2-loop
Using Kawai-Lewellen-Tye relation to SYM amp.
Only meta-stable vacuum
at finite N (flat directions)
negative specific heat !
small N, very low T regime
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

23. 1/N corrections (preliminary results)
Ishiki-Hanada-Hyakutake-J.N., work in progress
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

24. 5. Wilson loop and correlation functions
Hanada-Miwa-J.N.-Takeuchi, PRL 102 (’09)
Hanada-J.N.-Sekino-Yoneya, arXiv: [hep-th]

25. 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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

26. Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...
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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

27. Prediction from gauge/gravity duality (III)
correlation functions in gauge theory
generating functional
operator-field correspondence
gauge gravity
Gubser-Klebanov-Polyakov-Witten relation (’98)
SUGRA action evaluated at the classical solution
with the boundary condition
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

28. Result: 2pt correlation functions
e.g.)
predicted from dual geometry by Sekino-Yoneya (’99)
using Gubser-Klebanov-Polyakov-Witten relation (’98)
power law behavior
reproduced from 1d SYM
Hanada-J.N.-Sekino-Yoneya,
arXiv: [hep-th]
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

29. 6. Extension to higher dimensions based on large-N reduction

30. Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...
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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

31. 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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

32. A novel large N reduction for SYM on (cont’d)
Ishiki-Ishii-Shimasaki
-Tsuchiya (’08)
SYM on
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

33. 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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

34. 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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

35. Result: circular Wilson loop
(preliminary) 1 2
work in progress
Honda-Ishiki-J.N.-Tsuchiya
strong coupling
weak coupling
all order
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

36. Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...
Chiral primary operators
e.g.)
2-pt function
large-N reduction
SUSY non-renormalization theorem
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

37. Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...
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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

38. 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 meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

39. 7. Summary and future prospects

40. Summary and future prospects
Monte Carlo simulations of
1d U(N) SYM with 16 supercharges
Precision test of the gauge/gravity duality from first principles (non-conformal case)
negative specific heat ?
Black hole thermodynamics including corrections
Schwarzschild radius from Wilson loop
GKP-Witten prescription for correlation functions
gauge/gravity duality seems to hold including
corrections
non-perturbative definition of superstring theory
on a particular curved background
1d U(N) SYM with 8 supercharge
E v.s. T relation shows clear power law (p=2); gravity dual ?
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...

41. Summary and future prospects
extension to models in higher dimensoins
1d SYM supercharges
SYM on
32 supercharges
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)
IKKT matrix model (Ishibashi-Kawai-Kitazawa-Tsuchiya ’96)
0d reduction of the 1d SYM with 16 superchages
Emergence of 4d space-time from non-perturbative superstring theory
Anagnostopoulos, Aoyama, Azuma, Hanada, J.N.
Lattice meeting
Jun Nishimura (KEK) Numerical tests of the gauge-gravity duality...