Rindler Holography

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Rindler Holography

Daniel Grumiller

Institute for Theoretical Physics TU Wien

Workshop on Topics in Three Dimensional Gravity Trieste, March 2016

based on work w. H. Afshar, S. Detournay, W. Merbis, (B. Oblak), A. Perez, D. Tempo, R. Troncoso

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Outline

Motivation

Near horizon boundary conditions

Soft Heisenberg hair

Soft hairy black hole entropy

Concluding comments

Daniel Grumiller —RindlerHolography 2/23

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Outline

Motivation

Concluding comments

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There is a well-known system with many microstates studied for a long time (recently with help of computers)

Go: ≈10¹⁷² microstates

(SGo≈396)

Daniel Grumiller —RindlerHolography Motivation 4/23

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Go: ≈10¹⁷² microstates

(SGo≈396)

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Go: ≈10¹⁷² microstates (SGo ≈396)

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Go: ≈10¹⁷² microstates (SGo ≈396)→ black holes more complicated!

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Black hole microstates Bekenstein–Hawking

SBH= A

4G_N[forM:e^S^BH ∼ O(e¹⁰⁷⁶)∼echess microstates]

I Motivation: microscopic understanding of generic black hole entropy

I Microstate counting from CFT₂ symmetries (Carlip, Strominger, ...) using Cardy formula

I Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT, ...) → see talk byStephane Detournay!

I Main idea: consider near horizon symmetries for non-extremal horizons

I Near horizon line-element with Rindler acceleration a: ds² =−2aρ dv²+ 2 dvdρ+γ² dϕ²+. . . Meaning of coordinates:

I ρ: radial direction (ρ= 0 is horizon)

I ϕ∼ϕ+ 2π: angular direction

I v: (advanced) time

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SBH= A 4GN

I Microstate counting from CFT2 symmetries (Carlip, Strominger, ...) using Cardy formula

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SBH= A 4GN

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SBH= A 4GN

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SBH= A 4GN

I Near horizon line-element with Rindler acceleration a:

ds² =−2aρ dv²+ 2 dvdρ+γ² dϕ²+. . . Meaning of coordinates:

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Choices

I Rindler acceleration: state-dependent or chemical potential?

I If state-dependent: need mechanism to fix scale

— suggestion in 1512.08233:

v∼v+ 2πL

Works technically (see talk byHamid Afshar), but physical interpretation difficult

I If : all states in theory have same (Unruh-)temperature (see talk by Miguel Pino)

TU = a 2π

I Work in 3d Einstein gravity in Chern–Simons formulation (see talk by Jorge Zanelli!)

I_CS=±X

±

k 4π

Z

hA^±∧dA^±+²₃A^±∧A^±∧A^±i

with sl(2)connectionsA^± andk=`/(4GN) with AdS radius `= 1

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Choices

I If state-dependent: need mechanism to fix scale

— suggestion in 1512.08233:

v ∼v+ 2πL

Recall scale invariance

a→λa ρ→λρ v→v/λ of Rindlermetric

ds²=−2aρ dv²+ 2 dvdρ+γ² dϕ²

TU = a 2π

ICS=±X

±

k 4π

Z

hA^±∧dA^±+²₃A^±∧A^±∧A^±i with sl(2)connectionsA^± andk=`/(4GN) with AdS radius `= 1

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Choices

I If state-dependent: need mechanism to fix scale — suggestion in 1512.08233:

v ∼v+ 2πL

Recall scale invariance

a→λa ρ→λρ v→v/λ of Rindlermetric

ds²=−2aρ dv²+ 2 dvdρ+γ² dϕ²

TU = a 2π

ICS=±X

±

k 4π

Z

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Choices

v ∼v+ 2πL

I If chemical potential: all states in theory have same (Unruh-)temperature (see talk by Miguel Pino)

T_U = a 2π

I_CS=±X

±

k 4π

Z

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Choices

v ∼v+ 2πL

I Ifchemical potential: all states in theory have same (Unruh-)temperature (see talk by Miguel Pino)

T_U = a 2π

We make this choice in this talk!

I_CS=±X

±

k 4π

Z

hA^±∧dA^±+²₃A^±∧A^±∧A^±i with sl(2)connectionsA^± andk=`/(4G_N) with AdS radius `= 1

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Choices

v ∼v+ 2πL

I Ifchemical potential: all states in theory have same (Unruh-)temperature (see talk by Miguel Pino)

T_U = a 2π

I_CS=±X

±

k 4π

Z

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Outline

Motivation

Concluding comments

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Diagonal gauge

Standard trick: partially fix gauge

A^±=b⁻¹_± (ρ) d+a±(x⁰, x¹) b±(ρ) with some group element b∈SL(2) depending on radius ρ Drop ±decorations in most of talk

Manifold topologically a cylinder or torus, with radial coordinate ρ and boundary coordinates (x⁰, x¹)∼(v, ϕ)

I Standard AdS₃ approach: highest weight gauge

a∼L++L(x⁰, x¹)L− b(ρ) = exp (ρL0) sl(2): [Ln, Lm] = (n−m)Ln+m, n, m=−1,0,1

I For near horizon purposes diagonal gauge useful: a∼J(x⁰, x¹)L0

I Precise boundary conditions (ζ: chemical potential): a= (J dϕ+ζ dv) L₀

andb= exp (¹_ζL₁)·exp (^ρ₂L−1). (assume constantζ for simplicity)

Daniel Grumiller —RindlerHolography Near horizon boundary conditions 8/23

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Diagonal gauge

A=b⁻¹(ρ) d+a(x⁰, x¹) b(ρ) with some group element b∈SL(2) depending on radius ρ

I Standard AdS3 approach: highest weight gauge

a∼L₊+L(x⁰, x¹)L− b(ρ) = exp (ρL₀) sl(2): [L_n, L_m] = (n−m)L_n+m, n, m=−1,0,1

I For near horizon purposes diagonal gauge useful: a∼J(x⁰, x¹)L₀

I Precise boundary conditions (ζ: chemical potential): a= (J dϕ+ζ dv) L0

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Diagonal gauge

I For near horizon purposes diagonal gauge useful:

a∼J(x⁰, x¹)L₀

I Precise boundary conditions (ζ: chemical potential): a= (J dϕ+ζ dv) L0

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Diagonal gauge

I For near horizon purposes diagonal gauge useful:

a∼J(x⁰, x¹)L₀

I Precise boundary conditions (ζ: chemical potential):

a= (J dϕ+ζ dv) L0

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Near horizon metric Using

g_µν = ¹₂

A⁺_µ −A⁻_µ

A⁺_ν −A⁻_ν

yields (f := 1 +ρ/(2a))

ds²=−2aρfdv²+ 2 dvdρ−2ωa⁻¹ dϕdρ + 4ωρfdvdϕ+

γ²+^2ρ_af(γ²−ω²) dϕ² state-dependent functions J^± =γ±ω, chemical potentials ζ^±=−a±Ω Neglecting rotation terms (ω= 0) yields Rindlerplus higher order terms:

ds²=−2aρ dv²+ 2 dvdρ+γ² dϕ²+. . . Comments:

I Recover desired near horizon metric

I Rindler acceleration aindeed state-independent

I Two state-dependent functions (γ,ω) as usual in 3d gravity

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g_µν = ¹₂

A⁺_µ −A⁻_µ

A⁺_ν −A⁻_ν yields (f := 1 +ρ/(2a))

γ²+^2ρ_af(γ²−ω²) dϕ² state-dependent functions J^± =γ±ω, chemical potentialsζ^±=−a±Ω For simplicity set Ω = 0and a= const.in metric above

EOM imply ∂_vJ^± =±∂_ϕζ^±; in this case ∂_vJ^± = 0

Neglecting rotation terms (ω = 0) yields Rindlerplus higher order terms:

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g_µν = ¹₂

A⁺_µ −A⁻_µ

A⁺_ν −A⁻_ν yields (f := 1 +ρ/(2a))

γ²+^2ρ_af(γ²−ω²) dϕ² state-dependent functions J^± =γ±ω, chemical potentialsζ^±=−a±Ω Neglecting rotation terms (ω= 0) yields Rindlerplus higher order terms:

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g_µν = ¹₂

A⁺_µ −A⁻_µ

A⁺_ν −A⁻_ν yields (f := 1 +ρ/(2a))

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g_µν = ¹₂

A⁺_µ −A⁻_µ

A⁺_ν −A⁻_ν yields (f := 1 +ρ/(2a))

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Canonical boundary charges

I Canonical boundary charges non-zero for large trafos that preserve boundary conditions

I Zero mode charges: mass and angular momentum

For covariant approach to boundary charges see e.g. talks byKamal Hajian, Ali Seraj, Hossein Yavartanoo

Background independent result for Chern–Simons yields

Q[η] = k 4π

I

dϕ η(ϕ)J(ϕ)

I Finite

I Integrable

I Conserved

I Non-trivial

Meaningful near horizon boundary conditions and non-trivial theory!

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I Zero mode charges: mass and angular momentum Background independent result for Chern–Simons yields

Q[η] = k 4π

I

dϕ η(ϕ)J(ϕ)

I Finite

I Integrable

I Conserved

I Non-trivial

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I Zero mode charges: mass and angular momentum Background independent result for Chern–Simons yields

Q[η] = k 4π

I

dϕ η(ϕ)J(ϕ)

I Finite

I Integrable

I Conserved

I Non-trivial

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Outline

Motivation

Concluding comments

Daniel Grumiller —RindlerHolography Soft Heisenberg hair 11/23

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Near horizon symmetry algebra

I Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos Most general trafo

δa= d+ [a, ] =O(δa)

that preserves our boundary conditions for constantζ given by =⁺L₊+ηL₀+⁻L−

with

∂vη= 0 implying

δJ =∂_ϕη

I Expand charges in Fourier modes J_n^±= k

4π I

dϕ e^inϕJ^±(ϕ)

I Near horizon symmetry algebra J_n^±, J_m^±

=±¹₂knδn+m,0

J_n⁺, J_m⁻

= 0 Two u(1)ˆ current algebras with non-zero levels

I Much simpler than CFT2, warped CFT2, Galilean CFT2, etc.

I Map

P0=J₀⁺+J₀⁻ Pn= _knⁱ (J_−n⁺ +J_−n⁻ ) ifn6= 0 Xn=J_n⁺−J_n⁻ yields Heisenberg algebra(with CasimirsX0,P0)

[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0 [X_n, P_m]=iδ_n,m ifn6= 0

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I Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos

4π I

dϕ e^inϕJ^±(ϕ)

=±¹₂knδn+m,0

J_n⁺, J_m⁻

I Map

P₀=J₀⁺+J₀⁻ P_n= _knⁱ (J_−n⁺ +J_−n⁻ ) ifn6= 0 X_n=J_n⁺−J_n⁻ yields Heisenberg algebra(with CasimirsX0,P0)

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4π I

dϕ e^inϕJ^±(ϕ)

=±¹₂knδn+m,0

J_n⁺, J_m⁻

I Map

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4π I

dϕ e^inϕJ^±(ϕ)

=±¹₂knδn+m,0

J_n⁺, J_m⁻

I Much simpler than CFT₂, warped CFT₂, Galilean CFT₂, etc.

I Map

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4π I

dϕ e^inϕJ^±(ϕ)

=±¹₂knδn+m,0

J_n⁺, J_m⁻

I Much simpler than CFT₂, warped CFT₂, Galilean CFT₂, etc.

I Map

[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0 [Xn, Pm]=iδn,m ifn6= 0

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Soft hair

I Vacuum descendants|ψ(q)i

|ψ(q)i ∼Y (J_−n⁺ +

i

)^m⁺ⁱ Y (J_−n⁻ −

i

)^m⁻ⁱ |0i

I Hamiltonian

H:=Q[^±|_∂_v] =aP₀ commutes with all generators ofalgebra

I Energy of vacuum descendants

E_ψ =hψ(q)|H|ψ(q)i=Evachψ(q)|ψ(q)i=Evac

same as energy of vacuum

I Same conclusion true for descendants of any state! Soft hair = zero energy excitations on horizon

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Soft hair

|ψ(q)i ∼Y (J_−n⁺ +

i

)^m⁺ⁱ Y (J_−n⁻ −

i

)^m⁻ⁱ |0i

I Hamiltonian

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Soft hair

|ψ(q)i ∼Y (J_−n⁺ +

i

)^m⁺ⁱ Y (J_−n⁻ −

i

)^m⁻ⁱ |0i

I Hamiltonian

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Soft hair

|ψ(q)i ∼Y (J_−n⁺ +

i

)^m⁺ⁱ Y (J_−n⁻ −

i

)^m⁻ⁱ |0i

I Hamiltonian

I Same conclusion true for descendants of any state!

Soft hair = zero energy excitations on horizon

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Soft hair

|ψ(q)i ∼Y (J_−n⁺ +

i

)^m⁺ⁱ Y (J_−n⁻ −

i

)^m⁻ⁱ |0i

I Hamiltonian

I Same conclusion true for descendants of any state!

Soft hair = zero energy excitations on horizon

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Outline

Motivation

Concluding comments

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Macroscopic entropy

I Zero-mode solutions with constant chemical potentials: BTZ J₀^±= ^k₂(r₊±r−)

I Generic soft hairy black holes (or “black flowers”) from softly boosting BTZ

I Soft hairy black holes remain regular and have same energy as BTZ (for other boundary conditions generically not true)

I Macroscopic entropy

S= = A 4GN I No contribution from soft hair charges

I

Before addressing microstates consider map to aymptotic variables

Daniel Grumiller —RindlerHolography Soft hairy black hole entropy 15/23

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Macroscopic entropy

I

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Macroscopic entropy

I

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Macroscopic entropy

S = 2π(J₀⁺+J₀⁻) = A 4GN

calculated directly in Chern–Simons formulation

I No contribution from soft hair charges

I

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Macroscopic entropy

S = 2π(J₀⁺+J₀⁻) = A 4GN I No contribution from soft hair charges

I

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Macroscopic entropy

I Zero-mode solutions with constant chemical potentials: BTZ J₀^±= ^k₂(r+±r−)

S = 2π(J₀⁺+J₀⁻) = A 4GN I No contribution from soft hair charges

I Suggestive that microstate counting should work

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Macroscopic entropy

I Zero-mode solutions with constant chemical potentials: BTZ J₀^±= ^k₂(r+±r−)

S =2π(J₀⁺+J₀⁻)= A 4GN I No contribution from soft hair charges

I Suggestive that microstate counting should work

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Map to asymptotic variables

I Usual asymptotic AdS3 connection with chemical potential µ:

Aˆ= ˆb⁻¹ d+ˆaˆb ˆa_ϕ =L1−¹₂LL−1

ˆb=e^ρL⁰ ˆa_t=µL1−µ⁰L0+ ¹₂µ⁰⁰−¹₂Lµ L−1

I Gauge trafoˆa=g⁻¹(d+a)g with

g= exp (xL1)·exp (−¹₂JL−1) where∂vx−ζx=µ andx⁰−Jx= 1

I Near horizon chemical potential transforms into combination of asymptotic charge and chemical potential!

µ⁰−Jµ=−ζ

I Asymptotic charges: twisted Sugawara construction (remember comment in talk by Gaston Giribet) with near horizon charges

L= ¹₂J²+J⁰

I Get Virasoro with non-zero central charge δL= 2Lε⁰+L⁰ε−ε⁰⁰⁰

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ˆb=e^ρL⁰ ˆa_t=µL1−µ⁰L0+ ¹₂µ⁰⁰−¹₂Lµ L−1 I Gauge trafoˆa=g⁻¹(d+a)g with

g= exp (xL1)·exp (−¹₂JL−1) where∂_vx−ζx=µ andx⁰−Jx= 1

µ⁰−Jµ=−ζ

L= ¹₂J²+J⁰

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µ⁰−Jµ=−ζ

L= ¹₂J²+J⁰

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µ⁰−Jµ=−ζ

L= ¹₂J²+J⁰

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µ⁰−Jµ=−ζ

L= ¹₂J²+J⁰

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Remarks on asymptotic and near horizon variables

I Asymptotic spin-2 currents fulfill Virasoro algebra, but charges obey stillHeisenberg algebra

δQ=− k 4π

I

dϕ ε δL=− k 4π

I

dϕ η δJ

Reason: asymptotic “chemical potentials”µdepend on near horizon charges J and chemical potentialsζ

I Our boundary conditions singled out: whole spectrum compatible with regularity

I For constant chemical potential ζ: regularity =holonomy condition µµ⁰⁰−¹₂µ⁰²−µ²L=−2π²/β²

Solved automatically from map to asymptotic observables; reminder: µ⁰−Jµ=−ζ L= ¹₂J²+J⁰

Near horizon boundary conditions natural for near horizon observer

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δQ=− k 4π

I

dϕ η δJ

Solved automatically from map to asymptotic observables; reminder: µ⁰−Jµ=−ζ L= ¹₂J²+J⁰

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δQ=− k 4π

I

dϕ η δJ

Solved automatically from map to asymptotic observables; reminder:

µ⁰−Jµ=−ζ L= ¹₂J²+J⁰

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δQ=− k 4π

I

dϕ η δJ

Solved automatically from map to asymptotic observables; reminder:

µ⁰−Jµ=−ζ L= ¹₂J²+J⁰

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Cardy counting

I Idea: use map to asymptotic observables to do standard Cardy counting

I Twisted Sugawara construction expanded in Fourier modes kLn=X

p∈Z

Jn−pJp+

I Starting fromHeisenberg algebra obtain semi-classically Virasoro algebra

[L_n, L_m] = (n−m)L_n+m+

I Usual Cardy formula yields Bekenstein–Hawking result SCardy = 2π

q

L⁺₀ + 2π q

L⁻₀ = 2π(J₀⁺+J₀⁻) = A

4G_N =SBH

Precise numerical factor intwist term crucial for correct results

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Cardy counting

p∈Z

Jn−pJp+iknJn

[L_n, L_m] = (n−m)L_n+m+¹₂k n³δ_n+m,₀

q

kL⁺₀ + 2π q

kL⁻₀ = 2π(J₀⁺+J₀⁻) = A

4G_N =SBH

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Cardy counting

p∈Z

Jn−pJp+iknJn

[Ln, Lm] = (n−m)Ln+m+¹₂k n³δn+m,0

q

kL⁺₀ + 2π q

kL⁻₀ = 2π(J₀⁺+J₀⁻) = A

4G_N =SBH

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Cardy counting

p∈Z

Jn−pJp+iknJn

[Ln, Lm] = (n−m)Ln+m+¹₂k n³δn+m,0 I Usual Cardy formula yields Bekenstein–Hawking result

SCardy = 2π q

kL⁺₀ + 2π q

kL⁻₀ = 2π(J₀⁺+J₀⁻) = A

4G_N =SBH

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Cardy counting

p∈Z

Jn−pJp+iknJn

[Ln, Lm] = (n−m)Ln+m+¹₂k n³δn+m,0 I Usual Cardy formula yields Bekenstein–Hawking result

SCardy = 2π q

kL⁺₀ + 2π q

kL⁻₀ = 2π(J₀⁺+J₀⁻) = A

4G_N =SBH

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Warped CFT counting

See talk byStephane Detournay

I Mapnear horizon algebraJ_n^±= ¹₂(J_n±K_n) Yn∼X

Jn−pKp Tn∼Jn

to centerless warped conformal algebra

[Y_n, Y_m] = (n−m)Y_n+m [Yn, Tm] =−mT_n+m [Tn, Tm] = 0

I Modular propertyZ(β, θ) = Tr (e^−βH+iθJ) =Z(2πβ/θ,−4π²/θ) (H =Q[∂v],J =Q[∂ϕ]) projects partition function to ground state for small imaginary θ(we need θ→0)

I Assuming J^vac= 0yields

S =βH =SBH

Hamiltonian H is product of BH entropy and Unruh temperature

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Warped CFT counting

See talk byStephane Detournay

I Mapnear horizon algebraJ_n^±= ¹₂(J_n±K_n) Yn∼X

Jn−pKp Tn∼Jn

to centerless warped conformal algebra

[Y_n, Y_m] = (n−m)Y_n+m [Yn, Tm] =−mT_n+m [Tn, Tm] = 0

I Modular propertyZ(β, θ) = Tr (e^−βH+iθJ) =Z(2πβ/θ,−4π²/θ) (H =Q[∂v],J =Q[∂ϕ]) projects partition function to ground state for small imaginary θ(we need θ→0)

I Assuming J^vac= 0yields

S =βH =SBH

Hamiltonian H is product of BH entropy and Unruh temperature