Thermodynamics of the BMN matrix model at strong coupling Miguel S. - - PowerPoint PPT Presentation

Thermodynamics of the BMN matrix model at strong coupling

Miguel S. Costa Faculdade de Ciências da Universidade do Porto

Crete Center for Theoretical Physics - Heraklion, May 2014

Works with L. Greenspan, J. Penedones and J. Santos

Motivation

• Would like example where computations in both sides are within reach

Test and understand the gauge/gravity duality with observables that are not protected by SUSY or can be computed using integrability. How does gravitation phenomena, like black holes, emerge from gauge theory side?

Idea: Study thermodynamics of black holes dual to Matrix Quantum Mechanics that can be simulated on a computer using Monte-Carlo methods.

• Gauge/gravity duality as definition of quantum gravity in AdS

Dual CFT is renormalizable and unitary. Problem: how to decode the hologram? Unfortunately field theory very difficult in region of interest for quantum gravity (strong coupled; classical gravity , expansion loop expansion).

≡

N → ∞ 1/N

The case of D0-branes

• Closed strings interact with D0-branes in flat space

D0-brane Open string Closed string

• Closed strings interact with geometry produced by D0-branes

Closed string D0-brane geometry

⇔

[Itzhaki, Maldacena, Sonnenschein, Yankielowicz ´98]

D0-branes: field theoretic description (matrix quantum mechanics)

SD0 = N 2λ Z dt Tr  (DtXi)2 + ΨαDtΨα + 1 2 ⇥ Xi, Xj⇤2 + iΨαγj

αβ[Ψβ, Xj]

• Xi ≡ SU(N) bosonic matrices (i = 1, . . . , 9)

Ψ ≡ SU(N) fermionic matrices (16 real components) Dt = ∂t − i[A, ] ≡ covariant derivative γi ≡ SO(9) gamma matrices

• γi, γj

= 2δij

SO(9) global symmetry

• ‘t Hooft coupling is dimensionfull (relevant)

λ = g2

Y MN =

gsN (2π)2l3

s

≡ mass3

λeff = λ E3

E → ∞ (UV ) ≡ weak coupling E → 0 (IR) ≡ strong coupling

• Dual 10D gravitational coupling

16πGNl−8

s

= (2π)11 (λl3

s)2

N 2

• Theory on Euclidean time circle with periodicity β = 1/T

Dimensionless temperature τ = T λ1/3

Low temperatures is strong coupling SD0 = N 2λ Z β dt Tr  · · ·

• Can put theory on a computer using Monte Carlo simulations, accessing in

particular strong coupled region.

Dimensionless mean energy ✏ N 2 = E N 21/3

[Catterall, Wiseman ´07,´08,´09; Anagnostopoulos et al ´07; Hanada et al ´08,´13]

D0-branes: gravitational description

• 11D SUGRA solution (near horizon geometry of non-extremal D0-brane)

ds2 = dr2 f(r) + r2dΩ2

8 +

✓R r ◆7 dz2 + f(r)dt ✓ 2dz − ⇣r0 R ⌘7 dt ◆

f(r) = 1 − ⇣r0 r ⌘7 , ✓ R `s ◆7 = 60⇡3gsN , ✓r0 `s ◆5 = 120⇡2 49 (2⇡gsN)

5 3 ⌧ 2

• Classical gravity domain (at horizon)

1

N − 10

21

N − 5

9

N − 5

6

IIA 11D SUGRA G-L instability Horizon at scale

lP

Horizon at scale

ls

l2

sR(r0) ⌧ 1 ) τ ⌧ 1

gseφ(r0) ⌧ 1 ) τ N − 10

21

τ = T/λ1/3

τ

• Standard gravitation thermodynamics

✏ N 2 = c1⌧

14 5

S = AH 4GN = d1N 2τ

9 5

d1 = 4

13 5 15 2 5

⇣π 7 ⌘ 14

5

c1 = 9 14 d1 (because dE = TdS)

• nly fixes next power

in expansion

τ

fixes both coefficient and power of correction

1/N 2

• corrections give next term in expansion, at large N

α0

τ

1 16πGN Z d10x√−g e2φ⇣ R + · · · + α03R4 + . . . ⌘ ⇒ S N 2 = d1τ

9 5

⇣ 1 + d2τ

9 5

⌘ [Hanada et al´08]

• corrections to 11D SUGRA give 1/N correction (solution purely gravitational)

lP

1 16πG11 Z d11x√−g ⇣ R + l6

P R4⌘

⇒ S N 2 = d1τ

9 5 + 1

N 2 d3τ − 3

5

[Hanada et al´13]

checked predicted negligible

• Low temperature expansion predicted from gravity

✏ N 2 = h c1⌧

14 5 + c2⌧ 23 5 + . . .

i + 1 N 2 h c3⌧

2 5 + . . .

i + . . .

[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, !=6 N=17, !=8 7.41T2.8 7.41T2.8-5.58T4.6

SUGRA SUGRA + corrections

Monte-Carlo simulation

• f MQM

✏/N 2

τ

α0

• Low temperature expansion predicted from quantum gravity

✏ N 2 = h c1⌧

14 5 + c2⌧ 23 5 + . . .

i + 1 N 2 h c3⌧

2 5 + . . .

i + . . .

checked negligible [Hanada, Hyakutake, Ishiki, Nishimura ´13] [Hyakutake ´13]

Monte-Carlo simulation

• f MQM

✏/N 2

τ

Today’s talk is not about D0-brane matrix model

• Instability corresponds to Hawking radiation of D0-branes. At large N this is

suppressed and black hole is stable (positive specific heat).

• Today’s talk is about BMN matrix model

Mass deformation resolves IR divergence - canonical ensemble well defined. Much richer thermodynamics with a 1st order phase transition (at large N there are two dimensionless parameters). [Berenstein, Maldacena, Nastase ´02]

• Caveat: canonical ensemble ill defined - IR divergences from flat directions

in D0-brane moduli space. This is suppressed at large N (metastable state), but it is a source of tension in Monte Carlo simulations

F(T, r) N 2 ∼ Ffinite(T) + 9 N ln r

[Catterall, Wiseman ´09]

BMN matrix model

Canonical ensemble is well defined. Still “easy” to simulate on a computer. In large N ‘t Hooft limit dimensionless coupling constant

λ = g2

YMN

µ3

S = SD0 − N 2 Z dt Tr µ2 32 (Xi)2 + µ2 62 (Xa)2 + µ 4 Ψα 123

αβ Ψβ + i2µ

3 ✏ijkXiXjXk

• Massive deformation of D0-brane MQM. Preserves SUSY but breaks

SO(9) → SO(6) × SO(3)

a = 4, . . . , 9 i = 1, 2, 3

Many vacua

Xa = 0

Xi = µ 3 Ji

[Ji, Jj] = iijkJk

N = mn

M5−brane vacua ≡ m → ∞, n fixed Xi ∼     n × n . . . n × n . . . . . . . . . . . . . . . . . . n × n            m times D2−brane vacua ≡ n → ∞, m fixed (decoupled) Focus on trivial vacuum (single M5-brane) that is SO(9) invariant

Xi = Xa = 0

Exponential growth of spectrum with energy Hagedorn transition

→

Tc µ = 1 12 log 3  1 + 265 34 λ − c λ2 + O(λ3)

• ≈ 0.076 + O(λ)

First-order phase transition at [Hadizadeh, Ramadanovic, Semenoff, Young ’04]

N → ∞

• Thermodynamics ( )

Dimensionless temperature ≡ T µ Dimensionless coupling ≡ λ = g2

Y MN

µ3

T µ

TH µ

λ = g2

YMN

µ3

Confined phase Deconfined phase

F = O(N 2)

F = O(N 0) ?

W E A K C O U P L I N G

S T R O N G C O U P L I N G

Start here

Today: strongly coupled limit Dual geometry is SO(9) invariant non-extremal D0-brane with deformation turned on

µ → 0 , T µ fixed and large

Gravitational dual

• At strong coupling, for large temperature, dual geometry is SO(9) invariant and is

approximately the non-extremal D0-brane solution

ds2 = dr2 f(r) + r2dΩ2

8 + R7

r7 dz2 + f(r)dt ✓ 2dz − r7 R7 dt ◆

dC = µ dt ∧ dx1 ∧ dx2 ∧ dx3

Need back-reaction to decrease temperature and study phase transition at strong coupling. In particular,

SO(9) → SO(6) × SO(3)

Non-normalizable mode responsible for massive deformation

• The different vacua of BMN matrix model correspond to the Lin-Maldacena

geometries and asymptote to the M-theory plane wave solution

ds2 = dxidxi + dxadxa + 2dtdz − ✓µ2 32 xixi + µ2 62 xaxa ◆ dt2

dC = µ dt ∧ dx1 ∧ dx2 ∧ dx3

ds2 = −A (1 − y7) y7 dη2 + T4 y7  dζ + Ω (1 − y7)dη y7 2 + 1 y2 " B (dy + Fdx)2 (1 − y7)y2 + T1 4dx2 2 − x2 + T2 x2(2 − x2)dΩ2

2 + T3 (1 − x2)2dΩ2 5

# C = (M dη + L dζ) ∧ d2Ω2 Tailored to numerical implementation (domain of unknown is the unit square; everything dimensionless)

y

1

Horizon

∞

S2

S5

collapses collapses

1

x

| {z }

dΩ2

8

if T1=T2=T3=1 is a angular coordinate on compact 8-dimensional space with topology

x

S8

x = 1 x = 0 S2 S5 pole equator are functions of and

A, B, F, T1, T2, T3, T4, Ω, M, L

x

y y is a radial coordinate from boundary ( ) to horizon ( )

y = 0 y = 1 y = 0 y = 1 S8 × S1 × S1 S8 × S1 boundary horizon

• Ansatz for 11D SUGRA

ζ ∼ ζ + 2π M-theory circle

| {z }

dΩ2

8

if T1=T2=T3=1 ds2 = −A (1 − y7) y7 dη2 + T4 y7  dζ + Ω (1 − y7)dη y7 2 + 1 y2 " B (dy + Fdx)2 (1 − y7)y2 + T1 4dx2 2 − x2 + T2 x2(2 − x2)dΩ2

2 + T3 (1 − x2)2dΩ2 5

# C = (M dη + L dζ) ∧ d2Ω2 x = 1 x = 0 S2 S5 pole equator y = 0 y = 1 S8 × S1 × S1 S8 × S1 boundary horizon

• Ansatz for 11D SUGRA

Non-extremal D0-brane solution corresponds to

A = B = T1 = T2 = T3 = T4 = Ω = 1 , F = M = L = 0 , β = 4π 7 (Euclidean time circle)

This scaling symmetry will be important later... and need to use scaling symmetry of 11D SUGRA action with ζ ∼ ζ + 2π → ζ ∼ ζ + 2πs0 ⇒ I → s0I gµν → s2gµν , Cµνρ → s3Cµνρ ⇒ I → s9I s0 = ✓ R r0 ◆ 7

2 gs`s

r0 s = r0

invariant tensor hamonic SO(6) × SO(3)

• Boundary conditions

y

1

Horizon

∞

S2

S5

collapses collapses

1

x

y = 0 At infinity ( ): A, B, T1, T2, T3, T4, Ω → 1 ,

F → 0

M → ˆ µ x3(2 − x2)

3 2

y3 , L → 3 2 ˆ µ y4x3(2 − x2)

3 2

Recall that C = (M dη + L dζ) ∧ d2Ω2

ˆ µ = 7 12π µ T

To obtain physical solution do again above scalings, then geometry has same asymptotics of non-extremal D0-brane with temperature T and mass deformation turned on. The only parameter is This important to thermodynamics, because we just learned that I = s9s0 16πGN ˆ I ⇣ µ T ⌘ = 15 28 ✓ 15 142π8 ◆ 2

5

N 2 ✓ T λ

1 3

◆ 9

5 ˆ

I ⇣ µ T ⌘ S = s9s0 4GN ˆ S ⇣ µ T ⌘ = 15π 7 ✓ 15 142π8 ◆ 2

5

N 2 ✓ T λ

1 3

◆ 9

5 ˆ

S ⇣ µ T ⌘ Regularity at the axis of symmetry: horizon ( ), pole ( ) and equator ( ). y = 1 x = 0 x = 1

S2

S5

(β = 4π/7)

Einstein-DeTurck equations

ξµ = gαβ ⇣ Γµ

αβ − ˜

Γµ

αβ

⌘

DeTurck term that makes Einstein equations elliptic Connection of reference metric

Rµν r(µξν) = 1 12 ✓ FµαβγF αβγ

µ

1 12gµνF 2 ◆

d(F) + 1 2F ∧ F = 0

F = dC

Our reference metric is the non-extremal D0-brane solution. With appropriate boundary conditions on the numerical solution so solution also solves Einstein equations. ξµ = 0 [Headrick, Kitchen, Wiseman ’09]

A Smarr formula (good to check numerics)

• Integrate over surface of constant time with

d ⇣ ? Kv ⌘ = 0

y1 < y < y2

0 = Z

Σ12

d ⇣ ? Kv ⌘ = Z

∂Σ12

?Kv = Z

H

?Kv − Z

y→0

?Kv

• Let (time translations generator), then Smarr formula relates horizon

v = ∂ ∂η

area to boundary data

7 2 ˆ S = Z

y→0

?Kv

• Let be a killing vector. From field equations it follows that

is a conserved antisymmetric tensor, i.e.

vµ

d ⇣ ? Kv ⌘ = 0

(Kv)µν = rµvν + 1 3F µναβvγCαβγ + 1 6v[µF ν]αβγCαβγ

The solution

Q1 = 1 + y4q1 , Q2 = 1 + y4q2 , Q3 = y5q3 , Q4 = 1 + y4q4 , Q5 = 1 + y4q5 , Q6 = 1 + y4q6 , Q7 = 1 + y4q7 , Q8 = 1 + y4q8 , Q9 = ˆ µ(1 + y + y2 + y3 + y4) + y4q9 , Q10 = ˆ µ + yq10

• Horizon area and shape

0.0 0.5 1.0 1.5 2.0 120 140 160 180 m ` S `

Horizon area

0.0 0.5 1.0 1.5 2.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 m ` R2êR5

Ratio of maximal radius of to

S2 S5

S = 15π 7 ✓ 15 142π8 ◆ 2

5

N 2 ✓ T λ

1 3

◆ 9

5 ˆ

S ⇣ µ T ⌘ After scaling symmetry to obtain physical metric: Ri = ai ✓ T λ

1 3

◆ 2

5 ˆ

Ri ⇣ µ T ⌘ Reproduces scalings predicted from strongly coupled low energy moduli estimate in [Wiseman ’13]

∆N =

• 1 −

AreaN AreaN+1

• log ∆N ≈ −2.5 − 0.75N

25 30 35 40 45 50 1034 1030 1026 1022 1018

• N

log χ ≈ −17.5 − 0.75N

χ = max p ξνξν

25 30 35 40 45 50 1024 1022 1020 1018 1016

• Χ

Numerical convergence

Discretize PDEs using a Chebyshev grid with N x N points. Derivatives are estimated using polynomial approximations that involve all points in the grid - spectral methods.

requires regularization easy to measure

Black hole thermodynamics

• From scaling symmetry we saw that

F(T, µ) = −c0 T

14 5 ˆ

I ⇣ µ T ⌘ , S(T, µ) = c0 14 5 T

9 5 ˆ

S ⇣ µ T ⌘

Therefore ratio of free energies and entropies

F(T, µ) F(T, 0) = ˆ I µ

T

• ˆ

I(0) ≡ f ⇣ µ T ⌘ , S(T, µ) S(T, 0) = ˆ S µ

T

• ˆ

S(0) ≡ s ⇣ µ T ⌘

✓∂F ∂T ◆

µ

= −S

F(T, 0) = −c1T

14 5

First law Free energy

✓ 1 − 5 14 ˆ µ ∂ ∂ˆ µ ◆ f(ˆ µ) = s(ˆ µ)

)

⇒

Analyticity

⇒

s(ˆ µ) =

∞

X

n=0

sn ˆ µn , f(ˆ µ) =

∞

X

n=0

14sn 14 − 5n ˆ µn

Critical temperature

T < Tc

[Lin, Maldacena ’05]

• Phase transition occurs when free energy changes sign, since for

geometry without horizon is favoured F ∼ O

• N 0

Tc µ = 7 12πˆ µc ≈ 0.091

Similar to Hawking-Page phase transition in AdS

0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 m ` f

= −c1T

14 5 f(ˆ

µ) F(T, µ) = F(T, 0)f(ˆ µ)

• Note that BH is thermodynamically stable

c = T ✓∂S ∂T ◆

µ ⇒

c S = 9 5 − ˆ µ ∂ ∂ˆ µ log s(ˆ µ) > 0

Phase diagram at large N

T µ

λ = g2

YMN

µ3

Confined phase Deconfined phase

F = O(N 2)

F = O(N 0) ?

W E A K C O U P L I N G

S T R O N G C O U P L I N G

0.076

0.091

Very similar to SYM on a 3-sphere (µ ≡ 1/R) [Aharony, Marsano, Minwalla, Papadodimas, van Raamsdonk ’03]

To preserve depends on ratio of radii

E6

E3

Xi

Xa

θ

sin θ = R5 R2 = ✓XaXa XiXi ◆1/2 SO(6) × SO(3)

Boundary data (preliminary)

• The 10 functions admit expansion

near the boundary

(y = 0)

Qi(x, y)

Qi(x, y) = X

j

yj ˜ Qj

i(x)

SO(9)

• Boundary metric has symmetry, so are harmonic functions on .

Thus we can classify the invariant perturbations according to

• spin. This helps to establish bulk field / operator correspondence.

SO(9)

S8

SO(6) × SO(3)

˜ Qj

i(x)

µ

Ê Ê Ê ‡ ‡ ‡ ‡ ‡

1 3 5 7 9 2 4 6 1 3 5 7 9 2 4 6

ρ2

ρ4

ρ6

spin l

n(l)

vector mode

v(x, y) = X

l

⇣ ρl yn(l) + µl y˜

n(l)⌘

vl(x) + back reaction

α5 α6

β1 β2 β3

zero modes

δ7, γ7

Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡

2 4 6 8 10 2 4 6 2 4 6 8 10 2 4 6

α4

spin l

n(l)

scalar mode

s(x, y) = X

l

⇣ αl yn(l) + βl y˜

n(l)⌘

sl(x) + back reaction

normalizable non-normalizable

O ∼ Tr ([Xi, Xj]XA1 . . . XAl) , l ≥ 1 odd O ∼ Tr (XA1 . . . XAl) , l ≥ 2 even

0.0 0.5 1.0 1.5 2.0

• 0.4
• 0.3
• 0.2
• 0.1

0.0 m ` a2 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 m ` a5 0.0 0.5 1.0 1.5 2.0

• 0.35
• 0.30
• 0.25
• 0.20
• 0.15
• 0.10
• 0.05

0.00 m ` g7 0.0 0.5 1.0 1.5 2.0

• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 m ` d7

• We got for ρ2, α5, γ7, δ7

ρ2

Numerics pass highly non-trival check! Smarr formula OK: 7 2 ˆ S = 16π5 735 (98 + 280γ7 + 63δ7 − 276α2ˆ µ)

• Confirm phase diagram with Monte-Carlo simulations of PWMM
• Study other observables (expectation values) - holographic renormalization
• Study dynamical stability of our BH
• Construct BH duals of other vacua (different horizon topology)

(caveat: we really only determined upper limit on critical temperature)

• Deeper question: What makes the PWMM special?

What are the minimal ingredients of a quantum mechanical system such that it gives rise to classical gravity in the limit of many degrees of freedom? Future work

THANK YOU