Black holes and quantum gravity from super Yang-Mills...SYM in 10-d. g YM is dimensional; 2 YM = 3 p. We will consider 3, as then the SYM has it UV complete. Physics dual to gravity

Black holes and quantum gravity from superYang-Mills

Toby Wiseman (Imperial)

Kyoto ’15Numerical approaches to the holographic principle, quantum

gravity and cosmology

Plan

Introduction• Quantum gravity, black holes, large N SYM and numerical

methods

Previous meetings - London ’09, Santa Barbara ’12

• Black hole thermodynamics and the (non) lattice

More recent progress

• Progress in sugra• Progress on the lattice• The YM moduli

Dp-branes and SYM

Large N gauge theory provides the fascinating prospect of providinga description of quantum gravity. There are different proposals for thisthat we will hear more about. I will focus on the ’AdS/CFT’correspondence and its generalizations.

• There are other closely related proposals for the emergence ofgeometry (and cosmology) from large N matrix theories - see forexample the talks of Goro Ishiki and Asato Tsuchiya.

Dp-branes and SYM

Claim: Maximally supersymmetric (p + 1)-dimensional SU(N)Yang-Mills theory is physically equivalent to the full quantum stringtheory description of the decoupled dynamics of N Dp-branes.[ Maldacena ’98, Itzhaki et al ’98 - also BFSS and IKKT ’96 ]

• Vacuum at large N in string theory is an extremal black hole.• Such black holes have an ’infinite throat’.• Physics associated to ’decoupling’ is far down the throat of this

black hole.• It is ’low energy’ in the sense that is corresponds to highly

redshifted physics.• Includes all supergravity sector of the string theory - both

perturbative gravitons and non-perturbative physics such asblack holes.

Dp-branes and SYM

• Consider (p + 1)-d maximally susy Yang-Mills at large N;

LYM =1

g2YM

Tr[

14

F 2

µν+

12

DµΦIDµΦI − 14

[ΦI ,ΦJ

]2]

+ fermions

• N × N Hermitian matrix fields ΦI , where I = 1, . . . ,9− p

• May be thought of as classical dimensional reduction of N = 1SYM in 10-d.

• gYM is dimensional;[g2

YM

]= 3− p. We will consider p ≤ 3, as

then the SYM has it UV complete.• Physics dual to gravity requires large N; natural coupling isλ = Ng2

YM . Then also requires strongly coupling.

Dp-branes and SYM

The key questions in quantum gravity concern black holes. Whataccounts for their entropy, and how do they behave dynamically?

Thermodynamics

• At finite temperature T , the effective dimensionless coupling isλeff = λT p−3

• Also dimensionless temperature t = Tλ−1

3−p = λ− 1

3−p

eff .• We can hope to solve the SYM at finite temperature and

reproduce the behaviour of quantum black holes.

Dp-branes and SYM

The key questions in quantum gravity concern black holes. Whataccounts for their entropy, and how do they behave dynamically?

Dynamics

• How do black holes evapourate and encode their information inthe outgoing Hawking radiation? SYM is unitary, so there shouldbe no fundamental information loss?

• How do black holes form and thermalize?• Is the spacetime near the horizon smooth?

Obviously in recent years this has been envigorated by the extensivediscussions on ’firewalls’.

Dp-branes and SYM

There has been little progress analytically on either topic.

• It seems reasonable to think numerical methods may be the bestway to tackle these strongly coupled QFTs in the future. Thiswas the thinking behind the meetings in London and SantaBarabara, and now Kyoto.

• Bringing to together experts in quantum gravity, string theory andlattice/numerical QFT methods may provide powerful newpossibilities to answer many old and very fundamental questions.

Dp-branes and SYM

For the remainder of this talk I will consider the ’simpler’ problem ofdirectly simulating black holes in thermal equilibrium.

Aims• To test the holographic conjecture in a non-trivial setting• To learn new things about black holes and non-perturbative

quantum gravity

Later David Berenstein will discuss numerical approaches tosimulating time dependence for black holes.

Dp-branes and SYM

Perturbation theory

• When λeff 1 (t 1) then we may use PT finding;

ε ∼ N2T p+1

• Going to strong coupling we may use a supergravity dual whichpredicts a completely different behaviour...

Dp-branes and SYMSupergravity dual

• Gravity predicts, [ Gibbons, Maeda ’88 ; Garfinkle, Horowitz,Strominger ’91 ; Horowitz, Strominger ’91]

ε = (9− p)(

231−5p

(7−p)3(7−p)

) 15−p

N2

π11−2p

(Tλ−

13−p

)7−p

Ω(8−p)

25−p

λ1+p3−p

∼ N2t2(7−p)

5−p λ1+p3−p

• Gravity requires large N →∞ or get stringy (gs) corrections• Also require strong coupling λeff 1 (or t 1) or else α′

corrections.• However also ’stringy’ corrections if λeff too big (temp too small);

λeff O(

N2(5−p)

7−p

)

p = 0: quantum black holes in quantum mechanics

• I believe this is the simplest setting in which to study quantumgravity.

• All the interesting questions about quantum gravity are encodedin this theory,

• Have no fermion doubling and a trivial continuum.

Analytic attempts

The cases of p = 0 is particularly attractive as it is simply a ’quantummechanics problem’. There has been an interesting approach usingthe ’Gaussian approximation’.

• Gravity prediction;

ε

N2λ13

=2

315 × 7−

215 × 9× π 22

5

Ω25(8)

(Tλ−

13

) 145 ' 7.41t2.8

• Initial work by [ Kabat, Lifshytz, Lowe ’00-’01 ]

• More recent developments in [ Lin, Shao, Wang, Yin ’13 ]

Past numerical attempts

• Earliest numerical works; [ Hiller, Lunin, Pinsky, Trittmann, ’01;Wosiek, Campostrini ’04 ] However these studied correlationfunctions or hamiltonian - not easy to extract non-perturbativegravitational physics.

• The first work on the thermal problem using Euclidean numericalapproaches began in 2007 in the case p = 0 following earlierwork on the quenched system [ Aharony, Marsano, Minwalla, TW’04 ]

Past numerical attempts

Thermal/Euclidean approach

• Lattice approach [ Catterall, TW ] - utilizes the fact thatsupersymmetry is restored even using a naive Wilsondiscretization.

• Non-lattice approach [ Anagnostopoulos, Hanada, Nishimura,Takeuchi ] - utilizes the fact that one may fix a gauge up to theoverall Polyakov loop. The resulting action may then be Fourierdecomposed, which may give better convergence to thecontinuum. Again supersymmetry is thought to be restored in thecontinuum.

Lattice approach

SU(5) data from Catterall, TW

Non-lattice approach

Data from Anagnostopoulos et al

4

at the leading order was fitted nicely to the power lawE/N2 = 3.2 · T 2.7 within 0.25 ! T ! 1. Their resultsare in reasonable agreement with our data at T ∼ 1, butdisagree at lower temperature.

0

5

10

15

20

25

30

0.0 1.0 2.0 3.0 4.0 5.0

E/N

2

T

N=8, Λ=2N=12,Λ=4N=14, Λ=4black hole

HTE

0.8

1.0

1.2

0.45 0.500.8

1.0

1.2

0.45 0.50

FIG. 3: The energy is plotted against temperature. Thedashed line represents the result obtained by HTE up to thenext leading order for N = 8 [24]. The solid line representsthe energy predicted at small T by the gauge/gravity duality.The upper left panel zooms up the region, where the power-law behavior sets in.

Summary.— In this paper we have presented the firstMonte Carlo results for the maximally supersymmetricmatrix quantum mechanics, which is expected to play avery important role in string/M theories. The recentlyproposed non-lattice simulation together with the RHMCalgorithm enabled us to study the low temperature be-havior, which was not accessible by the high tempera-ture expansion. As we lower the temperature, we ob-served the infrared instability, which was found to beeliminated, however, by increasing N . We gave a naturalexplanation to this phenomenon based on the one-loopeffective action. Our data for the internal energy asymp-tote nicely to the result obtained from the dual geom-etry, which we consider as a highly nontrivial evidencefor the gauge/gravity duality in the non-conformal case.In particular, our results suggest that the maximally su-persymmetric matrix quantum mechanics exactly repro-duces not only the power but also the coefficient of thepower-law behavior obtained from the dual black-holegeometry.

Acknowledgments.— The authors would like to thankShoji Hashimoto and Hideo Matsufuru for helpful sugges-tions concerning the RHMC simulation. The computa-tions were carried out on supercomputers (SR11000 atKEK, SX8 at RCNP and SX7 at RIKEN) as well as onPC clusters. This work is supported by the EPEAEKprogrammes “Pythagoras II” and co-funded by the Eu-ropean Union (75%) and the Hellenic state (25%).

∗ Electronic address: [email protected]† Electronic address: [email protected]‡ Electronic address: [email protected]§ Electronic address: [email protected]

[1] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).[2] N. Itzhaki, J. M. Maldacena, J. Sonnenschein and

S. Yankielowicz, Phys. Rev. D 58, 046004 (1998).[3] J. L. F. Barbon, I. I. Kogan and E. Rabinovici, Nucl.

Phys. B 544, 104 (1999).[4] D. Kabat, G. Lifschytz and D. A. Lowe, Phys. Rev. Lett.

86, 1426 (2001); Phys. Rev. D 64, 124015 (2001).[5] O. Aharony, J. Marsano, S. Minwalla and T. Wiseman,

Class. Quant. Grav. 21, 5169 (2004).[6] O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas,

M. Van Raamsdonk and T. Wiseman, JHEP 0601, 140(2006).

[7] T. Banks, W. Fischler, S. H. Shenker and L. Susskind,Phys. Rev. D 55, 5112 (1997).

[8] E. Witten, Nucl. Phys. B 443, 85 (1995).[9] N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya,

Nucl. Phys. B 498, 467 (1997).[10] W. Krauth, H. Nicolai and M. Staudacher, Phys. Lett.

B 431, 31 (1998); W. Krauth and M. Staudacher, Phys.Lett. B 435, 350 (1998).

[11] T. Hotta, J. Nishimura and A. Tsuchiya, Nucl. Phys. B545, 543 (1999).

[12] Z. Burda, B. Petersson and J. Tabaczek, Nucl. Phys. B602, 399 (2001); Z. Burda, B. Petersson and M. Watten-berg, JHEP 0503, 058 (2005).

[13] J. Ambjorn, K. N. Anagnostopoulos, W. Bietenholz,T. Hotta and J. Nishimura, JHEP 0007, 011 (2000);JHEP 0007, 013 (2000).

[14] K. N. Anagnostopoulos and J. Nishimura, Phys. Rev. D66, 106008 (2002).

[15] H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada,Prog. Theor. Phys. 99, 713 (1999).

[16] J. Nishimura and F. Sugino, JHEP 0205, 001 (2002);H. Kawai, S. Kawamoto, T. Kuroki, T. Matsuo andS. Shinohara, Nucl. Phys. B 647, 153 (2002); T. Aoyamaand H. Kawai, Prog. Theor. Phys. 116, 405 (2006).

[17] M. Hanada, J. Nishimura and S. Takeuchi,arXiv:0706.1647 [hep-lat].

[18] S. Catterall and T. Wiseman, arXiv:0706.3518 [hep-lat].[19] M. A. Clark, A. D. Kennedy and Z. Sroczynski, Nucl.

Phys. Proc. Suppl. 140, 835 (2005).[20] S. Catterall and S. Karamov, Phys. Lett. B 528, 301

(2002).[21] M.A. Clark and A.D. Kennedy,

http://www.ph.ed.ac.uk/~mike/remez,2005.[22] B. Jegerlehner, arXiv:hep-lat/9612014.[23] P. Austing and J.F. Wheater, JHEP 0102, 028 (2001);

JHEP 0104, 019 (2001).[24] N. Kawahara, J. Nishimura and S. Takeuchi, in prepara-

tion.[25] R. A. Janik and J. Wosiek, Acta Phys. Polon. B 32 (2001)

2143.[26] N. Kawahara, J. Nishimura and S. Takeuchi,

arXiv:0706.3517 [hep-th].[27] I.R. Klebanov and A.A. Tseytlin, Nucl. Phys. B 475, 164

(1996).

Corrections to gravity

• Gravity prediction only holds for t 1 in N →∞ limit.

ε

N2λ13

= 7.41t2.8

• α′ corrections; [ Hanada, Hyakutake, Nishimura, Takeuchi ’08 ]

ε

N2λ13

= 7.41t2.8 + C t4.6 + . . .

• Quantum 1/N corrections; [ Hyakutake ’13 ] .

ε

N2λ13

= 7.41t2.8 − 5.771

N2 t0.4 + . . .

• While in gravity the computation of entropy requires only theclassical spacetime, it is important that its origin is fully quantum.

α′ corrections[ Hanada, Hyakutake, Nishimura, Takeuchi ’08 ]

4

If we instead make a one-parameter fit with p = 4.6 fixed,we obtain C = 5.58(1). This value, in turn, provides aprediction for the α′ corrections on the gravity side.

-4

-3

-2

-1

0

1

2

3

4

-1.0 -0.5 0.0 0.5

ln (7

.41T

2.8 -E

/N2 )

ln T

N=14, Λ=4N=17, Λ=6N=17, Λ=8

FIG. 1: The deviation of the internal energy 1N2 E from the

leading term 7.41 T145 is plotted against the temperature in

the log-log scale for λ = 1. The solid line represents a fitto a straight line with the slope 4.6 predicted from the α′

corrections on the gravity side.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

E/N

2

T

N=17, Λ=6N=17, Λ=8

7.41T2.8

7.41T2.8-5.58T4.6

FIG. 2: The internal energy 1N2 E is plotted against T for

λ = 1. The solid line represents the leading asymptotic be-havior at small T predicted by the gauge-gravity duality. Thedashed line represents a fit to the behavior (1) including thesubleading term with C = 5.58.

Summary.— We have discussed the α′ corrections tothe black hole thermodynamics, which enable us to de-termine the power of the sub-leading term in (1). Thispower is then found to be reproduced precisely by MonteCarlo data in gauge theory. Let us emphasize that thesubleading term is crucial for the precision test of thegauge-gravity duality. It is intriguing that our results ingauge theory can tell us the absence of O(α′) and O(α′2)

corrections to the supergravity action.

Recently [20] Monte Carlo data for the Wilson loopare also shown to reproduce a prediction obtained by es-timating the disk amplitude in the dual geometry. Unlikethe present case, α′ corrections to that quantity start atO(α′) due to the fluctuation of the string worldsheet andits coupling to the background dilaton field.

While it is certainly motivated to obtain the coefficientC of the subleading term from gravity, our results alreadyprovide a strong evidence that the gauge-gravity dualityholds including α′ corrections. This, in particular, im-plies that we can understand the microscopic origin ofthe black hole thermodynamics including α′ correctionsin terms of the open strings attached to the D0-branes.

Acknowledgments.— The authors would like to thankO. Aharony, K.N. Anagnostopoulos and A. Miwa for dis-cussions. The computations were carried out on super-computers (SR11000 at KEK) as well as on PC clustersat KEK and Yukawa Institute. The work of J.N. andY.H. is supported in part by Grant-in-Aid for ScientificResearch (Nos. 19340066, 20540286 and 19740141).

∗ Electronic address: [email protected]† Electronic address: [email protected]‡ Electronic address: [email protected]§ Electronic address: [email protected]

[1] T. Banks, W. Fischler, S. H. Shenker and L. Susskind,Phys. Rev. D 55, 5112 (1997).

[2] N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya,Nucl. Phys. B 498, 467 (1997).

[3] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).[4] N. Itzhaki, J. M. Maldacena, J. Sonnenschein and

S. Yankielowicz, Phys. Rev. D 58, 046004 (1998).[5] E. Witten, Nucl. Phys. B 443, 85 (1995).[6] K. N. Anagnostopoulos, M. Hanada, J. Nishimura and

S. Takeuchi, Phys. Rev. Lett. 100 (2008) 021601.[7] M. Hanada, J. Nishimura and S. Takeuchi, Phys. Rev.

Lett. 99, 161602 (2007).[8] N. Kawahara, J. Nishimura and S. Takeuchi, JHEP 0712

103, (2007).[9] S. Catterall and T. Wiseman, Phys. Rev. D 78, 041502

(2008); JHEP 0712, 104 (2007).[10] D. Kabat, G. Lifschytz and D. A. Lowe, Phys. Rev. Lett.

86, 1426 (2001); Phys. Rev. D 64, 124015 (2001).[11] I.R. Klebanov and A.A. Tseytlin, Nucl. Phys. B 475, 164

(1996).[12] D.J. Gross and E. Witten, Nucl. Phys. B 277, 1 (1986).[13] A.A. Tseytlin, Nucl. Phys. B 584, 233 (2000).[14] Y. Hyakutake and S. Ogushi, JHEP 0602, 068 (2006).[15] D.J. Gross and J.H. Sloan, Nucl. Phys. B 291, 41 (1987).[16] Y. Hyakutake, Prog. Theor. Phys. 118, 109 (2007).[17] G. Policastro and D. Tsimpis, Class. Quant. Grav. 23,

4753 (2006).[18] R.M. Wald, Phys. Rev. D 48 R3427 (1993).[19] V. Iyer and R.M. Wald, Phys. Rev. D 50 846 (1994).[20] M. Hanada, A. Miwa, J. Nishimura and S. Takeuchi,

arXiv:0811.2081.

Developments since Santa BarbaraNice results on finite N corrections [ Hanada, Hyakutake, Nishimura,Takeuchi ; Hyakutake ’13 ] .

(As we mentioned earlier, a is obtained as a = 5.58(1) by fitting the results from the gaugetheory side [22] in the temperature regime 0.5 T 0.7.) Therefore, we can actually testeq. (3) directly. In Fig. 4 we plot (Egauge Egravity)/N

2 against 1/N4 for T = 0.08 andT = 0.11. Our data are nicely fitted by straight lines passing through the origin. Thisimplies that our result obtained on the gauge theory side is indeed consistent with theresult (3) obtained on the gravity side including quantum gravity corrections. In the smallbox of the same figure, we plot Egauge/N

2 against 1/N2. The curves represent the fits tothe behavior Egauge/N

2 = 7.41 T 2.8 5.77 T 0.4/N2 + const./N4 expected from the gravityside. We find that the O(1/N4) term is comparable to the O(1/N2) term. The fact thatthe O(1/N6) term is not visible from our data is therefore quite nontrivial and worth beingunderstood from the gravity side. The agreement of similar accuracy is observed at othervalues of T .

Figure 4: The di↵erence (Egauge Egravity)/N2 as a function of 1/N4. We show the

results for T = 0.08 (squares) and T = 0.11 (circles). The data points can be nicely fittedby straight lines passing through the origin for each T . In the small box, we plot Egauge/N

2

against 1/N2 for T = 0.08 and T = 0.11. The curves represent the fits to the behaviorEgauge/N

2 = 7.41 T 2.8 5.77 T 0.4/N2 + const./N4 expected from the gravity side.

As a further consistency check, we have also fitted our results for each T by Egauge/N2 =

7.41 T 2.8 + c1/N2 + c2/N

4 leaving c1 and c2 as fitting parameters. In Fig. 5, we plot c1

obtained by the two-parameter fit against T , which agrees well with c1 = 5.77 T 0.4.As for the coecient c2 of the O(1/N4) terms, the prediction from the gravity side is

given by c2 = c T2.6 + · · · , where c is an unknown constant. In fact c2 can be fitted, for

10

Developments since Santa Barbara

Very nice new lattice data [ Kadoh, Kamata ’15 ; Filev, O’Connor ’15 ] .

0

5

10

15

20

25

30

0 1 2 3 4 5

E/N2

T

N=14N=32

GravityHTE(NLO,N=14)

Figure 3. Internal energy of the black hole against temperature. The simulation results (red for

N = 14, green for N = 32) coincide with the result of the high temperature expansion (dashed

orange curve) at high temperature and approach the theoretical prediction (dashed blue curve) as

the temperature decreases.

the theoretical prediction of the gravity side at the leading order of the expansion, (5.1) with

c2 = 0. The slightly curved orange line is the result of the high temperature expansion

[26], [27]. As can be seen in the figure, the lattice data coincide with the result of the

high temperature expansion at high temperature, while as the temperature decreases, they

smoothly approach the theoretical prediction of the gravity side.

In figure 4, we focus on low temperature of figure 3. The data surely approach the pre-

diction (dashed blue curve) and are likely to coincide with it as the temperature decreases

further. But, unfortunately, the temperatures we used in the simulations were not low

enough to explain the leading behavior of the gravity side. To obtain quantitative results

for the leading-order term, simulations at further low temperatures are required.

Instead, one can study the contribution of the next-to-leading order term by fitting

the lattice results using the following formula,

f(x) = 7.41x2.8 + Cxp, (5.5)

where C and p are the fitting parameters. From (5.1), if the duality conjecture is really

true, the obtained p should be 4.6 within the statistical errors. We performed the fit using

the 5 points within 0.375 ≤ T ≤ 0.475 and obtained

C = 9.0 ± 2.6, p = 4.74 ± 0.35. (5.6)

– 15 –

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9E/N2

T

N=14N=32

GravityNLO Fit

Figure 4. Low temperature region of the internal energy of the black hole. The dashed blue curve

is again the theoretical prediction of the gravity side up to the leading order. The dashed curve

denotes the fit result which is obtained by fitting 5 points within T = 0.375 − 0.475.

The obtained p is consistent with the theoretical prediction of the gravity side within

about seven percent statistical error. This is the first lattice result of the NLO term, which

quantitatively shows the validity of the duality conjecture in this system.

In [9], the NLO term was estimated from the numerical simulation based on the momen-

tum sharp cut-off method using the same fit formula (5.5). The fitted values C = 5.55(7),

p = 4.58(3) are consistent with our results (5.6) within two sigma. However, those values

were obtained from their data in a little high temperature region 0.5 ≤ T ≤ 0.7. We

also tried to fit the lattice results in the same temperature region, but could not obtain

a reasonable result within χ2/dof ! 1. This observation raised the possibility that the

temperature region 0.5 ≤ T ≤ 0.7 was a little high to estimate the next-leading order term.

The reason of the discrepancy between the two results have not been clear in detail so far.

Further simulations are now in progress and will give us the final answer.

6 Summary

Lattice gauge theory is a promising framework to reveal the gauge/gravity duality and the

quantum effects of gravity from the gauge theory side. In this paper, we have investigated

the duality from the lattice simulations of supersymmetric SU(N) Yang-Mills theory in

1+0 dimension with sixteen supercharges. The numerical results of the SUSY WTI have

shown that the Sugino lattice action that we used reproduces the correct continuum theory

in the continuum limit. We have also estimated the internal energy of the black hole and

– 16 –

Subtleties

There are however subtleties [ Catterall, TW ’09 ]

• Sign problem• Ill defined nature of the canonical partition function in low

dimensions.

Sign problem

In fact the sign problem doesn’t seem to be a problem at all. Studiedin [ Catterall, TW ’09; Catterall, Galvez, Joseph, Mehta ’11; Filev, O’Connor’15 ]

m=0.05

SUH3LSUH5L

0.0 0.5 1.0 1.5 2.0 2.5 3.0t0.90

0.92

0.94

0.96

0.98

1.00

cosHPfL

m=0.1

SUH3LSUH5L

0.0 0.5 1.0 1.5 2.0 2.5 3.0t0.80

0.85

0.90

0.95

1.00

cosHPfL

m=0.2

SUH3LSUH5L

0.0 0.5 1.0 1.5 2.0 2.5 3.0t0.70

0.75

0.80

0.85

0.90

0.95

1.00

cosHPfL

Figure 5: Plot of the average cosine of the Pfaan phase for N = 3 and 5 for 5 lattice points

for regulator masses m = 0.05 (top), 0.1 (middle) and 0.2 (bottom). We note that at low and high

temperatures the Pfaan phase appears to play little dynamical role. Its e↵ect appears to increase

as the regulator is removed, and as N is increased, although we always observe it to be rather small.

Reweighting other observables shown in later plots for N = 3, 5 gives no discernable di↵erence from

the phase quenched approximation.

to lend further weight to the claim, arguing that the thermal partition function is divergent

for any temperature and N , and being careful to consider the nature of the leading quantum

corrections to the classical moduli space from all fluctuations. Whilst a thermal potential

– 24 –

Divergence - moduli

[ Catterall, TW ’09 ]

• The classical vacua are gauge equivalent to configurationswhere Aµ and ΦI are both constant and diagonal.

• Coulomb branch; breaks U(N)→ U(1)N

• Due to the maximal susy this moduli space is robust to quantumcorrections (unlike in pure YM).

• While it is lifted by thermal effects, these can be controlled, andwe proved for p = 0 that this moduli space leads to a divergentpartition function.

• At finite N any bound state of branes is at best metastable.• This is associated to the fact that these black holes want to

radiate their constituent D-branes. See [ Lin, Shai, Wang, Xi ’13 ]for calculation of rate in gravity.

Divergence

See this ‘radiation’ in Monte-Carlosaw no eigenvalue divergence [12].

0 200 400 600 800 1000 1200

1

2

3

4

5

0 100 200 300 400 500

1

2

3

4

5

Figure 2: Plot of the maximum absolute value of the scalar eigenvalues against Monte Carlo configu-

ration number, for N = 5 with 5 lattice points with the top plot having t = 1.12 and the bottom plot

having t = 0.94. These represent typical Monte Carlo sequences and show that for lower temperatures

the divergence quickly sets in, while for higher temperatures it may take many configurations before

the instability is seen, with t ' 1.0 marking the divide in behaviour.

It is interesting that for high temperatures, one may use the Monte Carlo to sample the

strongly coupled region of the partition function for many configurations before one encounters

the divergent region and the Monte Carlo breaks down. This is essentially what allowed the

previous studies [12, 13] to display data for the thermal model without explicitly regulating

the divergence. In figure 3 we use this ‘metastability’ to plot the energy over temperature for

the unregulated theory in the phase quenched approximation. The solid small data points

– 19 –

BMN plane wave matrix model[ Berenstein, Maldacena, Nastase ’02 ]

Resolution proposed in [ Catterall, TW ’09 ]

• This divergence must be regulated.• There is an important modification of the system that cures the

instability - a supersymmetric mass term may be added thatpreserves all 16 supersymmetries.

• Mass term breaks R-symmetry from SO(9)→ SO(3)× SO(6).• This acts to lift the moduli space, even at the classical level.• The theory still has a gravity dual. The dual vacuum geometries

have less symmetry and are known ( [ Lin, Lunin, Maldacena ’04 ] ).• However the finite temperature black holes are very complicated;

there is a new dimensionless coupling - µ = mBMN/T .

• First simulations performed in [ Catterall, Van Anders ’10 ]

Development since Santa Barbara

New numerical GR methods required [ Headrick, Kitchen, TW ’09 ]BMN model dual black holes have now been found [ Costa,Greenspan, Penedones, Santos ’14 ]

smaller and dominate the thermal ensemble. In other words, the critical temperature for the

phase transition is 11

Tc

µ=

7

12bµc

= 0.105905(57) . (76)

-

μ

Figure 9: The free energy ratio f(bµ) obtained numerically using (75).

Let us now consider thermodynamical stability. The specific heat of the system is given

by

c = T

@S

@T

µ

. (77)

From (70) and (71) we may also express the specific heat in terms of the function s(bµ) as

c

S=

9

5 bµ @

@bµ log s(bµ) . (78)

Since in the range the black hole geometry is thermodynamically favoured, s(bµ) is a decreas-

ing function, as shown in Fig. 7b, we conclude that the specific heat is always positive and

therefore our solution is thermodynamically stable in this range.

4 Discussion

Our main result is the construction of the black hole geometry dual to the deconfined phase

of the PWMM. This allowed us to determine the value of the critical temperature at strong

coupling as depicted in the phase diagram 1. In addition, we determine the thermal expec-

tation values of several observables in the deconfined phase (see Fig. 8).

11We present the critical temperature with 6 digits because our numerical solutions satisfied the Smarrformulas with 106% accuracy and the polynomial fit in (75) decreases precision by one order of magnitude.

26

where f (µ) = F (µ)F (0) . To do: Reproduce this on the (non-) lattice!

Beyond p = 0

p = 1

• Only simulations studying gravity remain [ Catterall, Joseph, TW’10 ] where evidence for phase transition dual toGregory-Laflamme was seen.

• More accurate simulations are required; see recent [ Giguere,Kadoh ’15 ]

p = 3

• Susy lattice approach; [ Catterall, Damgaard, DeGrand, Galvez,Mehta ’12; Catterall, Damgaard, DeGrand, Giedt, Schaich ’14 ]

• Large N-equivalence with BMN quantum mechanics; [ Honda,Ishiki, Kim, Nishimura, Tsuchiya ’13 ]

• Will be interesting to see calculations relevant fornon-perturbative gravity.

Understanding black holes in SYM

• Work in [ TW ’13, Morita, Shiba, TW, Withers ’13, ’14 ]

• Attempt to have simple model for what is happening ‘inside’ theSYM

Moduli theory

• Recall we have classical vacua which are gauge equivalent toconfigurations where Aµ and ΦI are both constant and diagonal.

• Promote to classical moduli space (focus here on the scalarmoduli) ;

(ΦI)ab = φIa(xµ)δab

• The N moduli fields live on spacetime (functions of xµ) and arevalued in R9−p, the transverse space to the N Dp-branes.

• Denote with vector index; φIa → ~φa.

• The classical moduli action is;

Scl =Nλ

∫dτdxp

∑

a

12∂µ~φa · ∂µ~φa

Moduli theory

Quantum and thermal corrections• Moduli are weakly coupled far out on the Coulomb branch.• Define ’separation’ of two moduli, ~φa and ~φb in R9−p;

|φab|2 = (~φa − ~φb) · (~φa − ~φb)

• The moduli theory is weakly coupled when all branes are wellseparated.

• But does not correspond to strongly coupled gravity physics.• Instead reproduces the dynamics of branes which weakly

interact gravitationally.

Moduli theoryQuantum correction

• Classical action is corrected by loops from off-diagonal modes.Quantum corrections (arising at zero temperature) take the form,(using notation ~φab = ~φa − ~φb);[ BFSS ’96; Douglas, Kabat, Pouliot, Shenker ’96; Maldacena ’97 ; Douglas, Taylor ’98 ;

Jevicki, Yoneya ’98 ; Das ’99 ]

Squantum1-loop =− (2π)4−p

8(7− p)Ω8−p

×∫

dτdpx∑

a<b

2(∂µ~φab · ∂ν ~φab

)2−(∂µ~φab · ∂µ~φab

)2

∣∣∣~φab

∣∣∣7−p + . . .

• The dots . . . are higher derivative terms down by (∂φ)2/φ4.• Maximal susy implies first correction is at 4-derivative order.• This term is protected by supersymmetry. [ Dine, Seiberg ’97; Becker2 et

al ’97; Buchbinder et al ’99 ]

Moduli theory

Thermal correction• The potential receives a thermal correction [ TW ’13 ] ;

Sthermal1-loop = − 16

(2π)p/2

∫dτdpx

∑

a<b

UaU?b + UbU?

a

β1+p e−β|φab| (β|φab|)p/2

where Ua is the Polyakov loop around the Euclidean time circle.[ cf. Ambjorn, Makeenko, Semenoff ’98 ]

• Suppressed by the Boltzmann factor exp (−β|φab|).

Moduli theory at strong coupling

Claim• Estimate when moduli theory becomes strongly coupled by

equating the classical and leading 1-loop corrections in thesense of the virial thm.[ cf. Horowitz,Martinec; Li ’97; BFKS ’98 and more recently Smilga ’09 ]

• Ignore higher derivative and thermal corrections - checkconsistency after.

• Will reproduce features of the black brane thermodynamics.

ε = (9− p)(

231−5p

(7−p)3(7−p)

) 15−p

N2(π11−2pt7−p

Ω(8−p)

) 25−p

λ1+p3−p

Moduli theory at strong coupling

Estimates• Assume gross properties of thermal state controlled by one

physical scale χ. Estimate a thermal vev replacing φ as;

~φa ∼ ~φa − ~φb ∼ χ

• Derivatives are estimated using the thermal scale,

∂µ ∼ π T

This form assumes scaling - hence the derivatives are controlledonly by the thermal scale.

• Note the π is naturally associated with T and we wish to keeptrack of transcendental factors; e.g. for a Matsubara mode;

ψn(τ) = e2πniTτ , ∂τψn = 2πTniψn

Moduli theory at strong coupling

Estimates• Hence we make a replacement

~φa ∼ ~φa − ~φb ∼ χ

and

∂µ~φa ∼ ∂µ(~φa − ~φb

)∼ π T χ

• Finally we approximate the large N sums in the obvious way;∑

a

∼ N ,∑

a<b

∼ N2

Moduli theory at strong coupling

Estimates• Consider the SYM Euclidean action density L. In the moduli

approximation;

< L >'< Lcl > + < L1−loop >

where we will keep only the leading 1-loop term so;

Lcl =Nλ

N∑

a=1

12∂µ~φa · ∂µ~φa,

L1−loop,leading =(2π)4−p

8(7− p)Ω8−p

∑

a<b

2(∂µ~φab · ∂ν ~φab

)2−∣∣∣∂µ~φab

∣∣∣4

|φab|7−p

Moduli theory at strong coupling

Estimating Lcl

• Consider first the vev < Lcl >;

〈Lcl〉 =

⟨Nλ

N∑

a=1

12∂µ~φa · ∂µ~φa

⟩

∼ Nλ× N × (πTχ)2 =

N2π2T 2χ2

λ

Moduli theory at strong coupling

Estimating L1−loop

• Now consider the vev of the leading 1-loop term;

〈L1−loop〉 =

⟨(2π)4−p

8(7− p)Ω8−p

∑

a<b

2(∂µ~φab · ∂ν ~φab

)2−∣∣∣∂µ~φab

∣∣∣4

|φab|7−p

⟩

∼ π4−p

Ω8−p× N2 × (πTχ)4

χ7−p

=N2π8−pT 4

Ω8−pχ3−p

Moduli theory at strong coupling

Estimating strong coupling

• Close the estimates assuming the moduli theory is stronglycoupled;

〈Lcl〉 ∼ 〈L1−loop〉

• Why not factor of π? We may justify this from the virial theorem;⟨∫

dτdpx (2Lcl − (3− p)L1−loop)

⟩= 0

(may be viewed as the Schwinger-Dyson equation for thescaling; φI

a → (1 + ε)φIa, Aµa → (1 + ε)Aµa .)

Moduli theory at strong coupling

Estimating strong coupling

• Then 〈Lcl〉 ∼ 〈L1−loop〉 yields;

N2π2T 2χ2

λ∼ N2π8−pT 4

Ω8−pχ3−p =⇒ χ5−p ∼ λπ6−pT 2

Ω8−p

and fixes our physical scale χ.

Moduli theory at strong couplingEstimating the thermodynamics

• Consider the vev of the stress tensor of the SYM. Ignoring theindex structure we have;

〈Tµν〉 ∼ 〈Lcl〉 ∼N2π2T 2χ2

λ∼ N2π2T 2

λ

(λπ6−pT 2

Ω8−p

) 25−p

= N2

π11−2p

(Tλ−

13−p

)(7−p)

Ω8−p

25−p

λ1+p3−pp

• Recall from the black brane analysis;

ε = (9− p)(

231−5p

(7−p)3(7−p)

) 15−p

N2

π11−2p

(Tλ−

13−p

)7−p

Ω(8−p)

25−p

λ1+p3−p

Break down of thermodynamic prediction

Weak coupling and α′

• We dropped higher derivative 1-loop terms and thermal potentialterms in analysis.

• Can check self consistency of this and it precisely is consistent if1 λeff (or t 1)

• Can also see strong coupling corrections when λeff ∼ N2(5−p)

7−p .

Outlook• Unlike PT, extrapolating moduli theory to strong coupling does

yields interesting information about black holes• Current work with Morita and Berenstein to improve this physical

picture.• Can it be used to improve numerical simulation?

Outlook

Summary

• Very exciting time for numerical methods.• Highly non-trivial tests of holography have been performed.• However, the issue of regulating the divergence with BMN mass

is important.• Great potential for future simulations.

Goals• To extract new information about quantum gravity.• Almost certainly must better understand how local spacetime

emerges from SYM at large N.• To better understand analytic aspects of quantum black holes.

End of talk