Introduction to near horizon higher spin holography

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Introduction to near horizon higher spin holography

Daniel Grumiller

Institute for Theoretical Physics TU Wien

IIP programme ‘Strings at Dunes’

Natal, July 2016

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Simple punchline Heisenberg algebra

[X_n, P_m] =i δ_{n, m} fundamental not only in quantum mechanics

but also in near horizon physics of (higher spin) gravity theories based on work with

I Hamid Afshar

I Stephane Detournay

I Wout Merbis

I Blagoje Oblak

I Alfredo Perez

I Stefan Prohazka

I Shahin Sheikh-Jabbari

I David Tempo

I Ricardo Troncoso

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Outline

Motivation

Near horizon boundary conditions for spin-2

Generalization to spin-N

Soft Heisenberg hair

Soft hairy black hole entropy

Concluding comments

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Outline

Motivation

Concluding comments

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Black hole microstates Bekenstein–Hawking

SBH= A 4GN

I Motivation: microscopic understanding of generic black hole entropy

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

I Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT, ...)

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 (Strominger, Carlip, ...) using Cardy formula

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SBH= A 4GN

warped CFT: Detournay, Hartman, Hofman ’12

Galilean CFT: Bagchi, Detournay, Fareghbal, Simon ’13; Barnich ’13

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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 but physical interpretation difficult

I If : all states in theory have same (Unruh-)temperature T_U = a

2π

I Work in 3d Einstein gravity in Chern–Simons formulation 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

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

I If : all states in theory have same (Unruh-)temperature TU = a

2π

I Work in 3d Einstein gravity in Chern–Simons formulation 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

Works technically but physical interpretation difficult Recall scale invariance

a→λa ρ→λρ v→v/λ of Rindlermetric

ds²=−2aρ dv²+ 2 dvdρ+γ² dϕ²

I If : all states in theory have same (Unruh-)temperature TU = a

2π

I Work in 3d Einstein gravity in Chern–Simons formulation 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

T_U = a 2π

±

k 4π

Z

hA^±∧dA^±+²₃A^±∧A^±∧A^±i

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Choices

v ∼v+ 2πL

I Ifchemical potential: all states in theory have same (Unruh-)temperature

T_U = a 2π suggestion in1511.08687

We make this choice in this talk!

±

k 4π

Z

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Choices

v ∼v+ 2πL

I Ifchemical potential: all states in theory have same (Unruh-)temperature

T_U = a 2π

±

k 4π

Z

hA^±∧dA^±+²₃A^±∧A^±∧A^±i

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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 with δb= 0 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 δa=δJ dϕ L₀ andb= exp (¹_ζL₊)·exp (^ρ₂L−). (assume constantζ for simplicity)

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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 with δb= 0

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 δa=δJ dϕ L0

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

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Diagonal gauge

A=b⁻¹(ρ) d+a(x⁰, x¹) b(ρ)

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 δa=δJ dϕ L0

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Diagonal gauge

A=b⁻¹(ρ) d+a(x⁰, x¹) b(ρ)

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 δa=δJ dϕ 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

I γ =γ(ϕ): “black flower”

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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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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

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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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) What should we expect?

I Virasoro? (spacetime is locally AdS3)

I BMS3? (Rindler boundary similar to scri)

I warped conformal algebra? (this is what we found for Rindleresque holography and what Donnay, Giribet, Gonzalez, Pino found in their near horizon analysis)

=±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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Outline

Motivation

Concluding comments

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Higher spin near horizon boundary conditions in diagonal gauge

I Inspired by spin-2 take same group elementb in A=b⁻¹(ρ) d+a(v, ϕ)

b(ρ)and choose a=

N

X

s=2

J_(s)W₀^(s) dϕ+

N

X

s=2

ζ_(s)W₀^(s) dv with W_n⁽²⁾=L_n andJ_(s)=J

I Reminder: relevant part ofsl(N) algebra: [L_n, W_m^(s)] = n(s−1)−m

W_n+m^(s) so Wn^(s) are generators associated with spin-s

I EOM imply

∂_vJ_(s)=∂_ϕζ_(s)(= 0 in this talk)

I (non-trivial) boundary condition preserving trafos generated by =

N

X

s=2

η_(s)W₀^(s)

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b(ρ)and choose a=

N

X

s=2

J_(s)W₀^(s) dϕ+

N

X

s=2

I Reminder: relevant part ofsl(N) algebra:

[L_n, W_m^(s)] = n(s−1)−m W_n+m^(s) so Wn^(s) are generators associated with spin-s

I EOM imply

N

X

s=2

η_(s)W₀^(s)

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b(ρ)and choose a=

N

X

s=2

J_(s)W₀^(s) dϕ+

N

X

s=2

I EOM imply

N

X

s=2

η_(s)W₀^(s)

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b(ρ)and choose a=

N

X

s=2

J_(s)W₀^(s) dϕ+

N

X

s=2

I EOM imply

N

Xη_(s)W₀^(s)

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Higher spin near horizon symmetry algebra

I Construct again canonical charges Q[η]∼

I

dϕ η_(s)(ϕ)J_(s)(ϕ)

and introduce again Fourier componentsJ_(s)_n∼H

dϕ e^inϕJ_(s)(ϕ)

I Their algebra is again the Heisenberg algebra

[(X_s)_n,(X_t)_m] = [(P_s)_n,(P_t)_m] = [(X_s)₀,(P_t)_n] = [(P_s)₀,(X_t)_n] = 0 [(X_s)_m,(P_t)_n] =iδ_s,tδ_m,n for m6= 0.

with similar redefinitions as before (Ps)0 =J_{(s) 0}⁺ +J_{(s) 0}⁻ (P_s)_n∝ 1

n(J_(s)⁺ _−n) +J_(s)⁻ _−n) forn6= 0 (Xs)n=J_(s)⁺ _n−J_(s)⁻ _n

and2N −2 Casimirs(Xs)0,(Ps)0 with s= 2. . . N

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Higher spin near horizon symmetry algebra

I Construct again canonical charges Q[η]∼

I

dϕ η_(s)(ϕ)J_(s)(ϕ)

and introduce again Fourier componentsJ_(s)_n∼H

dϕ e^inϕJ_(s)(ϕ)

I Their algebra is again the Heisenberg algebra

[(X_s)_n,(X_t)_m] = [(P_s)_n,(P_t)_m] = [(X_s)₀,(P_t)_n] = [(P_s)₀,(X_t)_n] = 0 [(X_s)_m,(P_t)_n] =iδ_s,tδ_m,n for m6= 0.

with similar redefinitions as before (Ps)0 =J_{(s) 0}⁺ +J_{(s) 0}⁻ (P_s)_n∝ 1

n(J_(s)⁺ _−n) +J_(s)⁻ _−n) forn6= 0 (Xs)n=J_(s)⁺ _n−J_(s)⁻ _n

and2N −2 Casimirs(Xs)0,(Ps)0 with s= 2. . . N

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Outline

Motivation

Concluding comments

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Soft hair

I Vacuum spin-2 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

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

same as energy of vacuum

I Same conclusion true for (higher spin) 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

H:=Q[^±|_∂_v] =aP0

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Soft hair

|ψ(q)i ∼Y (J_−n⁺ +

i

)^m⁺ⁱ Y (J_−n⁻ −

i

)^m⁻ⁱ |0i

I Hamiltonian

H:=Q[^±|_∂_v] =aP0

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Soft hair

|ψ(q)i ∼Y (J_−n⁺ +

i

)^m⁺ⁱ Y (J_−n⁻ −

i

)^m⁻ⁱ |0i

I Hamiltonian

H:=Q[^±|_∂_v] =aP0

I Same conclusion true for (higher spin) 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

H:=Q[^±|_∂_v] =aP0

I Same conclusion true for (higher spin) 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^±∼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 =

I No contribution from soft hair charges or higher spin charges

I

Before addressing microstates consider map to aymptotic variables

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Macroscopic entropy

S =

I

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Macroscopic entropy

S =

I

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Macroscopic entropy

S = 2π(J₀⁺+J₀⁻)

calculated directly in Chern–Simons formulation (in spin-2 case:

S =A/(4G_N))

I

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Macroscopic entropy

S = 2π(J₀⁺+J₀⁻)

I

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Macroscopic entropy

I Zero-mode solutions with constant chemical potentials: BTZ J₀^±∼r+±r−

S = 2π(J₀⁺+J₀⁻)

I Suggestive that microstate counting should work

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Macroscopic entropy

I Zero-mode solutions with constant chemical potentials: BTZ J₀^±∼r+±r−

S =2π(J₀⁺+J₀⁻)

I Suggestive that microstate counting should work

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Map to asymptotic variables in spin-2 case

I Usual asymptotic AdS₃ connection with chemical potential µ:

Aˆ= ˆb⁻¹ d+ˆaˆb ˆa_ϕ=L+−¹₂LL−

ˆb=e^ρL⁰ ˆa_t=µL+−µ⁰L0+ ¹₂µ⁰⁰−¹₂Lµ L−

I Gauge trafoˆa=g⁻¹(d+a)g with

g= exp (xL₊)·exp (−¹₂JL−) 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 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=µL+−µ⁰L0+ ¹₂µ⁰⁰−¹₂Lµ L− I Gauge trafoˆa=g⁻¹(d+a)g with

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 in spin-2 case

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

I Twisted Sugawara construction expanded in Fourier modes kL_n=X

p∈Z

Jn−pJ_p+iknJ_n

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

I NeedJ₀^vac=ik/2 to get correct vacuum valueL^vac₀ =−k/4; with a=iget modular transformed line-element ds² =ρ²dt+ dρ²−dϕ² Precise numerical factor intwist term crucial for correct results