Black holes and quantum gravity from super Yang-Mills Toby Wiseman - PowerPoint PPT Presentation

Black holes and quantum gravity from super Yang-Mills Toby Wiseman (Imperial) Kyoto ’15 Numerical 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

D p -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.

D p -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 D p -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.

D p -branes and SYM • Consider ( p + 1 ) -d maximally susy Yang-Mills at large N ; � 1 Φ I , Φ J � 2 � � 1 µν + 1 2 D µ Φ I D µ Φ I − 1 4 F 2 L YM = + fermions Tr g 2 4 YM • 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 2 • g YM is dimensional; = 3 − p . We will consider p ≤ 3, as YM then the SYM has it UV complete. • Physics dual to gravity requires large N ; natural coupling is λ = Ng 2 YM . Then also requires strongly coupling .

D p -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 = λ T p − 3 1 1 − 3 − p = λ • Also dimensionless temperature t = T λ − 3 − p . eff • We can hope to solve the SYM at finite temperature and reproduce the behaviour of quantum black holes.

D p -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’.

D p -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.

D p -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.

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

D p -branes and SYM Supergravity dual • Gravity predicts, [ Gibbons, Maeda ’88 ; Garfinkle, Horowitz, Strominger ’91 ; Horowitz, Strominger ’91 ]   π 11 − 2 p � � 7 − p 2 5 − p 1 � � T λ − 3 − p 1 5 − p N 2   1 + p 2 31 − 5 p ǫ = ( 9 − p ) λ   3 − p ( 7 − p ) 3 ( 7 − p ) Ω ( 8 − p ) 2 ( 7 − p ) 1 + p N 2 t 5 − p λ ∼ 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); � � 2 ( 5 − p ) λ eff ≪ O N 7 − p

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.

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; � � 14 31 5 × 7 − 21 22 5 × 9 × π ǫ 2 5 ≃ 7 . 41 t 2 . 8 5 T λ − 1 = 3 1 2 N 2 λ Ω 3 5 ( 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 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 ]

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.

Lattice approach SU(5) data from Catterall, TW

Non-lattice approach Data from Anagnostopoulos et al 30 1.2 1.2 25 1.0 1.0 20 E/N 2 0.8 0.8 15 0.45 0.45 0.50 0.50 10 N=8, Λ =2 N=12, Λ =4 5 N=14, Λ =4 black hole HTE 0 0.0 1.0 2.0 3.0 4.0 5.0 T

Corrections to gravity • Gravity prediction only holds for t ≪ 1 in N → ∞ limit. ǫ 7 . 41 t 2 . 8 = 1 N 2 λ 3 • α ′ corrections; [ Hanada, Hyakutake, Nishimura, Takeuchi ’08 ] ǫ 7 . 41 t 2 . 8 + C t 4 . 6 + . . . = 1 N 2 λ 3 • Quantum 1 / N corrections; [ Hyakutake ’13 ] . ǫ 7 . 41 t 2 . 8 − 5 . 77 1 N 2 t 0 . 4 + . . . = 1 N 2 λ 3 • While in gravity the computation of entropy requires only the classical spacetime, it is important that its origin is fully quantum.

α ′ corrections [ Hanada, Hyakutake, Nishimura, Takeuchi ’08 ] 3.0 N=17, Λ =6 N=17, Λ =8 2.5 7.41T 2.8 7.41T 2.8 -5.58T 4.6 2.0 E/N 2 1.5 1.0 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 T FIG. 2: The internal energy N 2 E is plotted against T for 1 λ = 1. The solid line represents the leading asymptotic be- havior 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 = 5 . 58.

Developments since Santa Barbara Nice results on finite N corrections [ Hanada, Hyakutake, Nishimura, Takeuchi ; Hyakutake ’13 ] . The difference ( E gauge − E gravity ) /N 2 as a function of 1 /N 4 . We show the Figure 4: 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 2 . 8 − 5 . 77 T 0 . 4 /N 2 + const ./N 4 expected from the gravity side.

Developments since Santa Barbara Very nice new lattice data [ Kadoh, Kamata ’15 ; Filev, O’Connor ’15 ] . 30 3 N=14 25 N=32 2.5 Gravity NLO Fit 20 2 E/N 2 15 1.5 10 1 N=14 N=32 0.5 5 Gravity HTE(NLO,N=14) 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 2 3 4 5 T T

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. Studied in [ Catterall, TW ’09; Catterall, Galvez, Joseph, Mehta ’11; Filev, O’Connor ’15 ] cos H Pf L 1.00 0.98 0.96 m = 0.05 0.94 SU H 3 L SU H 5 L 0.92 0.90 3.0 t 0.0 0.5 1.0 1.5 2.0 2.5 cos H Pf L 1.00 0.95 0.90 0.1 H 3 L 0.85 H 5 L 0.80 3.0 t 0.0 0.5 1.0 1.5 2.0 2.5 H Pf L 1.00 0.95 0.90 0.85 0.2 H 3 L 0.80 H 5 L 0.75 0.70 3.0 t 0.0 0.5 1.0 1.5 2.0 2.5