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

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.

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.

Dp-branes and SYM

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

LYM = 1 g2

YM

Tr 1 4F 2

µν + 1

2DµΦIDµΦI − 1 4

• ΦI, ΦJ2

+ fermions

• N × N Hermitian matrix fields ΦI, where I = 1, . . . , 9 − p
• May be thought of as classical dimensional reduction of N = 1

SYM 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 is

λ = Ng2

• YM. Then also requires strongly coupling.

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

λeff = 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. 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’.

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.

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.

Dp-branes and SYM

Perturbation theory

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

ǫ ∼ N2T p+1

• Going to strong coupling we may use a supergravity dual which

predicts a completely different behaviour...

Dp-branes and SYM

Supergravity dual

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

Strominger ’91 ; Horowitz, Strominger ’91]

ǫ = (9 − p)

• 231−5p

(7−p)3(7−p)

• 1

5−p N2

   π11−2p Tλ−

1 3−p

7−p Ω(8−p)   

2 5−p

λ

1+p 3−p

∼ N2t

2(7−p) 5−p λ 1+p 3−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

• N

2(5−p) 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;

ǫ N2λ

1 3

= 2

31 5 × 7− 21 5 × 9 × π 22 5

Ω

2 5

(8)

• Tλ− 1

3

14

5 ≃ 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 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

• verall 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

5 10 15 20 25 30 0.0 1.0 2.0 3.0 4.0 5.0 E/N2 T N=8, Λ=2 N=12,Λ=4 N=14, Λ=4 black hole HTE 0.8 1.0 1.2 0.45 0.50 0.8 1.0 1.2 0.45 0.50

Corrections to gravity

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

ǫ N2λ

1 3

= 7.41t2.8

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

ǫ N2λ

1 3

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

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

ǫ N2λ

1 3

= 7.41t2.8 − 5.77 1 N2 t0.4 + . . .

• 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 ]

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/N2 T N=17, Λ=6 N=17, Λ=8 7.41T2.8 7.41T2.8-5.58T4.6

• FIG. 2:

The internal energy

1 N2 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. 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 ] .

Figure 4: The difference (Egauge − Egravity)/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 Egauge/N 2 against 1/N 2 for T = 0.08 and T = 0.11. The curves represent the fits to the behavior Egauge/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 ] .

5 10 15 20 25 30 1 2 3 4 5

E/N2 T

N=14 N=32 Gravity HTE(NLO,N=14) 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

T

N=14 N=32 Gravity NLO Fit

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 ]

m=0.05 SUH3L SUH5L 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t 0.90 0.92 0.94 0.96 0.98 1.00 cosHPfL 0.1 H3L H5L 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t 0.80 0.85 0.90 0.95 1.00 cosHPfL 0.2 H3L H5L 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t 0.70 0.75 0.80 0.85 0.90 0.95 1.00 HPfL

Divergence - moduli

[ Catterall, TW ’09 ]

• The classical vacua are gauge equivalent to configurations

where 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 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.

Divergence

See this ‘radiation’ in Monte-Carlo

200 400 600 800 1000 1200 1 2 3 4 5 100 200 300 400 500 1 2 3 4 5

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 = 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 ] b

• μ
• 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 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.

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

Moduli theory

• Recall we have classical vacua which are gauge equivalent to

configurations where Aµ and ΦI are both constant and diagonal.

• Promote to classical moduli space (focus here on the scalar

moduli) ; (ΦI)ab = φI

a(xµ)δab

• The N moduli fields live on spacetime (functions of xµ) and are

valued in R9−p, the transverse space to the N Dp-branes.

• Denote with vector index; φI

a →

φa.

• The classical moduli action is;

Scl = N λ

• dτdxp

a

1 2∂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 well

separated.

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

interact gravitationally.

Moduli theory

Quantum 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 ]

Squantum

1-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 ] ;

Sthermal

1-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 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.

ǫ = (9 − p)

• 231−5p

(7−p)3(7−p)

• 1

5−p N2

π11−2pt7−p Ω(8−p)

• 2

5−p

λ

1+p 3−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 controlled

• nly 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; 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 = N λ

N

• a=1

1 2∂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 =

• N

λ

N

• a=1

1 2∂µ φa · ∂µ φa

• ∼ N

λ × 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 strongly

coupled; 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 the scaling; φI

a → (1 + ǫ)φI a, Aµ a → (1 + ǫ)Aµ a.)

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 coupling

Estimating the thermodynamics

• Consider the vev of the stress tensor of the SYM. Ignoring the

index structure we have; Tµν ∼ Lcl ∼ N2π2T 2χ2 λ ∼ N2π2T 2 λ λπ6−pT 2 Ω8−p

• 2

5−p

= N2    π11−2p Tλ−

1 3−p

(7−p) Ω8−p   

2 5−p

λ

1+p 3−p

p

• Recall from the black brane analysis;

ǫ = (9 − p)

• 231−5p

(7−p)3(7−p)

• 1

5−p N2

   π11−2p Tλ−

1 3−p

7−p Ω(8−p)   

2 5−p

λ

1+p 3−p

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 ≪ λeff (or t ≪ 1)

• Can also see strong coupling corrections when λeff ∼ 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?

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