Black holes and quantum gravity from super Yang-Mills

Transcription

1 Black holes and quantum gravity from super Yang-Mills Toby Wiseman (Imperial) Kyoto 15 Numerical approaches to the holographic principle, quantum gravity and cosmology

2 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

3 Dp-branes and SYM Large N gauge theory provides the fascinating prospect of providing a description of quantum gravity. There are different proposals for this that 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 of geometry (and cosmology) from large N matrix theories - see for example the talks of Goro Ishiki and Asato Tsuchiya.

4 Dp-branes and SYM Claim: Maximally supersymmetric (p + 1)-dimensional SU(N) Yang-Mills theory is physically equivalent to the full quantum string theory 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 as black holes.

5 Dp-branes and SYM Consider (p + 1)-d maximally susy Yang-Mills at large N; L YM = 1 [ 1 gym 2 Tr 4 F µν 2 Dµ Φ I D µ Φ I 1 [ Φ I, Φ J] ] 2 + fermions 4 N N Hermitian matrix fields Φ I, where I = 1,..., 9 p May be thought of as classical dimensional reduction of N = 1 SYM in 10-d. g YM is dimensional; [ g 2 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 is λ = NgYM 2. Then also requires strongly coupling.

6 Dp-branes and SYM The key questions in quantum gravity concern black holes. What accounts for their entropy, and how do they behave dynamically? Thermodynamics At finite temperature T, the effective dimensionless coupling is λ eff = λt p 3 Also dimensionless temperature t = T λ 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.

7 Dp-branes and SYM The key questions in quantum gravity concern black holes. What accounts for their entropy, and how do they behave dynamically? Dynamics How do black holes evapourate and encode their information in the outgoing Hawking radiation? SYM is unitary, so there should be 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 extensive discussions on firewalls.

8 Dp-branes and SYM There has been little progress analytically on either topic. It seems reasonable to think numerical methods may be the best way to tackle these strongly coupled QFTs in the future. This was the thinking behind the meetings in London and Santa Barabara, and now Kyoto. Bringing to together experts in quantum gravity, string theory and lattice/numerical QFT methods may provide powerful new possibilities to answer many old and very fundamental questions.

9 Dp-branes and SYM For the remainder of this talk I will consider the simpler problem of directly 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 to simulating time dependence for black holes.

10 Dp-branes and SYM Perturbation theory When λ eff 1 (t 1) then we may use PT finding; ɛ N 2 T p+1 Going to strong coupling we may use a supergravity dual which predicts a completely different behaviour...

11 Supergravity dual Dp-branes and SYM Gravity predicts, [ Gibbons, Maeda 88 ; Garfinkle, Horowitz, Strominger 91 ; Horowitz, Strominger 91] ( ɛ = (9 p) ) p 1 (7 p) 3(7 p) 5 p N 2 ( π 11 2p T λ 1 3 p Ω (8 p) ) 7 p 2 5 p λ 1+p 3 p N 2 t 2(7 p) 5 p λ 1+p 3 p Gravity requires large N or get stringy (g s ) corrections Also require strong coupling λ eff 1 (or t 1) or else α corrections. However also stringy corrections if λ eff too big (temp too small); ) λ eff O (N 2(5 p) 7 p

12 p = 0: quantum black holes in quantum mechanics I believe this is the simplest setting in which to study quantum gravity. All the interesting questions about quantum gravity are encoded in this theory, Have no fermion doubling and a trivial continuum.

13 Analytic attempts The cases of p = 0 is particularly attractive as it is simply a quantum mechanics problem. There has been an interesting approach using the Gaussian approximation. Gravity prediction; ɛ N 2 λ 1 3 = π 22 5 Ω 2 5 (8) ( ) T λ t 2.8 Initial work by [ Kabat, Lifshytz, Lowe ] More recent developments in [ Lin, Shao, Wang, Yin 13 ]

14 Past numerical attempts Earliest numerical works; [ Hiller, Lunin, Pinsky, Trittmann, 01; Wosiek, Campostrini 04 ] However these studied correlation functions or hamiltonian - not easy to extract non-perturbative gravitational physics. The first work on the thermal problem using Euclidean numerical approaches began in 2007 in the case p = 0 following earlier work on the quenched system [ Aharony, Marsano, Minwalla, TW 04 ]

15 Past numerical attempts Thermal/Euclidean approach Lattice approach [ Catterall, TW ] - utilizes the fact that supersymmetry is restored even using a naive Wilson discretization. Non-lattice approach [ Anagnostopoulos, Hanada, Nishimura, Takeuchi ] - utilizes the fact that one may fix a gauge up to the overall Polyakov loop. The resulting action may then be Fourier decomposed, which may give better convergence to the continuum. Again supersymmetry is thought to be restored in the continuum.

16 SU(5) data from Catterall, TW Lattice approach

17 Non-lattice approach at the leading order was fitted nicely to the power law E/N 2 =3.2 T 2.7 within 0.25 T 1. Their results are in reasonable agreement with our data at T 1, but Data disagree from Anagnostopoulos at lower temperature. et al E/N N=8, Λ=2 N=12,Λ=4 5 N=14, Λ=4 black hole HTE FIG. 3: The energy is plotted against temperature. The dashed line represents the result obtained by HTE up to the T Electronic add Electronic add Electronic add Electronic add [1] J. M. Maldacen [2] N. Itzhaki, J S. Yankielowic [3] J. L. F. Barbo Phys. B 544, 1 [4] D. Kabat, G. L 86, 1426(2001 [5] O. Aharony, J Class. Quant. [6] O. Aharony, J M. Van Raam (2006). [7] T. Banks, W. Phys. Rev. D 5 [8] E. Witten, Nu [9] N. Ishibashi, H Nucl. Phys. B [10] W. Krauth, H B 431, 31(199

18 Corrections to gravity Gravity prediction only holds for t 1 in N limit. ɛ N 2 λ 1 3 = 7.41t 2.8 α corrections; [ Hanada, Hyakutake, Nishimura, Takeuchi 08 ] ɛ N 2 λ 1 3 = 7.41t C t Quantum 1/N corrections; [ Hyakutake 13 ]. ɛ N 2 λ 1 3 = 7.41t N 2 t While in gravity the computation of entropy requires only the classical spacetime, it is important that its origin is fully quantum.

19 leading term 7.41 T 14 5 is plotted against the temperature in the log-log scale for λ =1. Thesolidlinerepresentsafit to a straight line with the slope α 4.6 predicted from the α corrections corrections on the gravity side. [ Hanada, Hyakutake, Nishimura, Takeuchi 08 ] E/N N=17, Λ=6 N=17, Λ=8 7.41T T T FIG. 2: The internal energy 1 N 2 E is plotted against T for λ =1. Thesolidlinerepresentstheleadingasymptoticbehavior at small T predicted by the gauge-gravity duality. The dashed line represents a fit to the behavior (1) including the subleading term with C = T N Research (Nos , 205 Electronic address: masano Electronic address: hyakuta Electronic address: jnishi@p Electronic address: shingo@ [1] T. Banks, W. Fischler, S. H Phys. Rev. D 55, 5112(199 [2] N. Ishibashi, H. Kawai, Y. Nucl. Phys. B 498, 467(19 [3] J. M. Maldacena, Adv. Theo [4] N. Itzhaki, J. M. Malda S. Yankielowicz, Phys. Rev. [5] E. Witten, Nucl. Phys. B 4 [6] K. N. Anagnostopoulos, M S. Takeuchi, Phys. Rev. Let [7] M. Hanada, J. Nishimura a Lett. 99, (2007). [8] N. Kawahara, J. Nishimura 103, (2007). [9] S. Catterall and T. Wisema (2008); JHEP 0712, 104(2 [10] D. Kabat, G. Lifschytz and 86, 1426(2001);Phys.Rev [11] I.R. Klebanov and A.A. Tse (1996). [12] D.J. Gross and E. Witten, N [13] A.A. Tseytlin, Nucl. Phys.

20 implies that our result obtained on the gauge theory side is indeed consistent with the result (3) obtained on the gravity side including quantum gravity corrections. In the small box of the same figure, we plot E gauge /N 2 against 1/N 2. The curves represent the fits to 5.77 T Developments 0.4 /N since 2 +const./n Santa 4 expected from the gravity Barbara the behavior E gauge /N 2 =7.41 T 2.8 side. We find that the O(1/N 4 )termiscomparabletotheo(1/n 2 )term. Thefactthat the O(1/N 6 )termisnotvisiblefromourdataisthereforequitenontrivialandworthbeing Nice understood results fromon thefinite gravity side. N corrections The agreement of[ similar Hanada, accuracy Hyakutake, is observed at Nishimura, other Takeuchi values of T ;. Hyakutake 13 ]. Figure 4: The di erence (E gauge E gravity )/N 2 as a function of 1/N 4. We show the results for T =0.08 (squares) and T =0.11 (circles). The data points can be nicely fitted by straight lines passing through the origin for each T. In the small box, we plot E gauge /N 2 against 1/N 2 for T =0.08 and T =0.11. The curves represent the fits to the behavior E gauge /N 2 =7.41 T T 0.4 /N 2 +const./n 4 expected from the gravity side.

21 The obtained p is consistent with the theoretical prediction ofthegravityside about seven percent statistical error. This is the first lattice result of the NLO term quantitatively shows the validity of the duality conjecture inthissystem. Developments since Santa Barbara Very nice new lattice data [ Kadoh, Kamata 15 ; Filev, O Connor 15 ] N=14 N=32 Gravity NLO Fit E/N 2 15 E/N N=14 N=32 Gravity HTE(NLO,N=14) T T igure 3. Internal energy of the black hole against temperature. Figure The simulation 4. Low temperature results (redregion for of the internal energy of the black hole. The dashed blu = 14, greenforn = 32) coincide with the result of the highistemperature again the theoretical expansion prediction (dashed of the gravity side up to the leading order. Thedashe ange curve) at high temperature and approach the theoretical prediction denotes the (dashed fit result blue which curve) is obtained as by fitting 5 points within T = e temperature decreases.

22 Subtleties There are however subtleties [ Catterall, TW 09 ] Sign problem Ill defined nature of the canonical partition function in low dimensions.

23 Sign problem In fact the sign problem doesn t seem to be a problem at all. Studied in [ Catterall, TW 09; Catterall, Galvez, Joseph, Mehta 11; Filev, O Connor 15 ] coshpfl m= SUH3L SUH5L t coshpfl

24 Divergence - moduli [ Catterall, TW 09 ] The classical vacua are gauge equivalent to configurations where A µ and Φ I are both constant and diagonal. Coulomb branch; breaks U(N) U(1) N Due to the maximal susy this moduli space is robust to quantum corrections (unlike in pure YM). While it is lifted by thermal effects, these can be controlled, and we proved for p = 0 that this moduli space leads to a divergent partition 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.

25 Divergence saw no eigenvalue divergence [12]. See this radiation in Monte-Carlo

26 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 that preserves 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 = m BMN /T. First simulations performed in [ Catterall, Van Anders 10 ]

27 Development since Santa Barbara te the thermal ensemble. In other words, the critical temperature for the 11 NewTnumerical c GR methods required [ Headrick, Kitchen, TW 09 ] BMNµ model = 7 = (57). (76) 12 bµ dual c black holes have now been found [ Costa, Greenspan, Penedones, Santos 14 ] - μ : The free where energy f (µ) ratio = f(bµ) F (µ) F (0). obtained To do: numerically Reproduce using(75). this on the (non-) lattice! sider thermodynamical stability. The specific heat of the system is given

28 Beyond p = 0 p = 1 Only simulations studying gravity remain [ Catterall, Joseph, TW 10 ] where evidence for phase transition dual to Gregory-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 for non-perturbative gravity.

29 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 the SYM

30 Moduli theory Recall we have classical vacua which are gauge equivalent to configurations where A µ and Φ I are both constant and diagonal. Promote to classical moduli space (focus here on the scalar moduli) ; (Φ I ) ab = φ I a(x µ )δ ab The N moduli fields live on spacetime (functions of x µ ) and are valued in R 9 p, the transverse space to the N Dp-branes. Denote with vector index; φ I a φ a. The classical moduli action is; S cl = N dτdx p λ a 1 2 µ φa µ φa

31 Moduli theory Quantum and thermal corrections Moduli are weakly coupled far out on the Coulomb branch. Define separation of two moduli, φ a and φ b in R 9 p ; φ ab 2 = ( φ a φ b ) ( φ a φ b ) The moduli theory is weakly coupled when all branes are well separated. But does not correspond to strongly coupled gravity physics. Instead reproduces the dynamics of branes which weakly interact gravitationally.

32 Quantum correction Moduli theory Classical action is corrected by loops from off-diagonal modes. Quantum corrections (arising at zero temperature) take the form, (using notation φ ab = φ a φ b ); [ BFSS 96; Douglas, Kabat, Pouliot, Shenker 96; Maldacena 97 ; Douglas, Taylor 98 ; Jevicki, Yoneya 98 ; Das 99 ] S quantum 1-loop (2π) 4 p = 8(7 p)ω 8 p ) 2 ) 2 dτd p x 2 ( µ φab νφab ( µ φab µ φab a<b φ p ab The dots... are higher derivative terms down by ( φ) 2 /φ 4. Maximal susy implies first correction is at 4-derivative order. This term is protected by supersymmetry. [ Dine, Seiberg 97; Becker 2 et al 97; Buchbinder et al 99 ]

33 Moduli theory Thermal correction The potential receives a thermal correction [ TW 13 ] ; S1-loop thermal = 16 dτd p x U a Ub + U bua (2π) p/2 β 1+p e β φab (β φ ab ) p/2 a<b where U a is the Polyakov loop around the Euclidean time circle. [ cf. Ambjorn, Makeenko, Semenoff 98 ] Suppressed by the Boltzmann factor exp ( β φ ab ).

34 Moduli theory at strong coupling Claim Estimate when moduli theory becomes strongly coupled by equating the classical and leading 1-loop corrections in the sense of the virial thm. [ cf. Horowitz,Martinec; Li 97; BFKS 98 and more recently Smilga 09 ] Ignore higher derivative and thermal corrections - check consistency after. Will reproduce features of the black brane thermodynamics. ( ɛ = (9 p) ) p 1 (7 p) 3(7 p) 5 p N 2 ( π 11 2p t 7 p Ω (8 p) ) 2 5 p λ 1+p 3 p

35 Estimates Moduli theory at strong coupling Assume gross properties of thermal state controlled by one physical scale χ. Estimate a thermal vev replacing φ as; φ a φ a φ b χ Derivatives are estimated using the thermal scale, µ π T This form assumes scaling - hence the derivatives are controlled only by the thermal scale. Note the π is naturally associated with T and we wish to keep track of transcendental factors; e.g. for a Matsubara mode; ψ n (τ) = e 2πniT τ, τ ψ n = 2πTniψ n

36 Moduli theory at strong coupling Estimates Hence we make a replacement φ a φ a φ b χ and µ φa µ ( φa φ b ) π T χ Finally we approximate the large N sums in the obvious way; N, N 2 a a<b

37 Moduli theory at strong coupling Estimates Consider the SYM Euclidean action density L. In the moduli approximation; < L > < L cl > + < L 1 loop > where we will keep only the leading 1-loop term so; L cl = N λ L 1 loop,leading = N a=1 1 2 µ φa µ φa, (2π) 4 p 8(7 p)ω 8 p a<b ) 2 2 ( µ φab νφab µ φab 4 φ ab 7 p

38 Moduli theory at strong coupling Estimating L cl Consider first the vev < L cl >; L cl = N λ N a=1 1 2 µ φa µ φa N λ N (πt χ)2 = N2 π 2 T 2 χ 2 λ

39 Moduli theory at strong coupling Estimating L 1 loop Now consider the vev of the leading 1-loop term; ) 2 (2π) 4 p 2 ( µ φab ν µ φab 4 φab L 1 loop = 8(7 p)ω 8 p φ ab 7 p = N2 π 8 p T 4 Ω 8 p χ 3 p a<b π 4 p Ω 8 p N 2 (πt χ) 4 χ 7 p

40 Moduli theory at strong coupling Estimating strong coupling Close the estimates assuming the moduli theory is strongly coupled; L cl L 1 loop Why not factor of π? We may justify this from the virial theorem; dτd p x (2L cl (3 p)l 1 loop ) = 0 (may be viewed as the Schwinger-Dyson equation for the scaling; φ I a (1 + ɛ)φ I a, A µ a (1 + ɛ)a µ a.)

41 Moduli theory at strong coupling Estimating strong coupling Then L cl L 1 loop yields; N 2 π 2 T 2 χ 2 λ N2 π 8 p T 4 Ω 8 p χ 3 p = χ 5 p λπ6 p T 2 Ω 8 p and fixes our physical scale χ.

42 Moduli theory at strong coupling Estimating the thermodynamics Consider the vev of the stress tensor of the SYM. Ignoring the index structure we have; T µν L cl N2 π 2 T 2 χ 2 N2 π 2 T 2 ( λπ 6 p T 2 λ λ Ω 8 p ( ) (7 p) 2 π 11 2p T λ 1 5 p 3 p = N 2 Ω 8 p ) 2 5 p 1+p 3 p λp Recall from the black brane analysis; ( ɛ = (9 p) ) p 1 (7 p) 3(7 p) 5 p N 2 ( π 11 2p T λ 1 3 p Ω (8 p) ) 7 p 2 5 p λ 1+p 3 p

43 Break down of thermodynamic prediction Weak coupling and α We dropped higher derivative 1-loop terms and thermal potential terms in analysis. Can check self consistency of this and it precisely is consistent if 1 λ eff (or t 1) Can also see strong coupling corrections when λ eff N 2(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?

44 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.

45 End of talk