Near horizon dynamics of three dimensional black holes

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Near horizon dynamics of three dimensional black holes

Daniel Grumiller

Institute for Theoretical Physics TU Wien

Seminar talk at ICTS, Bangalore, August, 2019

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Outline

Overture

Hamiltonian reduction

Near horizon boundary conditions

Near horizon Hamiltonian

KdV deformation

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Outline

Overture

KdV deformation

Conclusions

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Main message

I Near horizon boundary action for 3-dimensional black holes SNH[Φ⁺,Φ⁻] =

Z

dtdσ Π⁺Φ˙⁺+ Π⁻Φ˙⁻− H_NH(Φ⁺,Φ⁻)

I Scalar fields Φ^± denote left/right movers along the horizon I Scalar fields are self-dual (Floreanini–Jackiw-like)

Π∼Φ⁰

I Near horizon Hamilton density is total derivative HNH(Φ)∼ζΦ⁰

Manifestation of “softness” of near horizon excitations

Purpose of talk: explain and derive results summarized above

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Main message

Z

dtdσ Π⁺Φ˙⁺+ Π⁻Φ˙⁻− H_NH(Φ⁺,Φ⁻) I Scalar fields Φ^± denote left/right movers along the horizon

I Scalar fields are self-dual (Floreanini–Jackiw-like) Π∼Φ⁰

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Main message

Z

dtdσ Π⁺Φ˙⁺+ Π⁻Φ˙⁻− H_NH(Φ⁺,Φ⁻) I Scalar fields Φ^± denote left/right movers along the horizon

to reduce clutter: drop ±decorations in rest of talk

I Scalar fields are self-dual (Floreanini–Jackiw-like) Π∼Φ⁰

I Near horizon Hamilton density is total derivative H_NH(Φ)∼ζΦ⁰

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Main message

Z

dtdσ Π⁺Φ˙⁺+ Π⁻Φ˙⁻− H_NH(Φ⁺,Φ⁻) I Scalar fields Φ^± denote left/right movers along the horizon I Scalar fields are self-dual (Floreanini–Jackiw-like)

Π∼Φ⁰

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Main message

Z

Π∼Φ⁰

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Main message

Z

Π∼Φ⁰

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Outline

Overture

KdV deformation

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Einstein gravity in three dimensions as Chern–Simons theory Einstein gravity in three dimensions useful toy model:

IEH3[g] = 1 16πG

Z

M

d³x√

−g R+ 2

`²

+ ˆI∂M

I no local physical degrees of freedom ⇒ simple!

I rotating (BTZ) black hole solutions analogous to Kerr ds² =−(r²−r₊²)(r²−r²₋)

`²r² dt²+ `²r²dr²

(r²−r₊²)(r²−r₋²)+r²

dϕ−r+r−

`r² dt 2

I Brown–Henneaux asymptotic symmetries: 2 Virasoros (AdS3/CFT2) [L_n, L_m] = (n−m)L_n+m+ c

12(n³−n)δ_n+m,₀ c= 3` 2G

= 6k

I Gauge theoretic formulation as Chern–Simons theory [k=`/(4G)] ICS[A] = k

4π Z

M

Tr A∧dA+²₃A∧A∧A +I∂M

SO(2,2)connectionAusually split into two SL(2,R)connections; drop all±decorations & work with single sector

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IEH3[g] = 1 16πG

Z

M

d³x√

−g R+ 2

`²

+ ˆI∂M

`²r² dt²+ `²r²dr²

(r²−r₊²)(r²−r₋²)+r²

dϕ−r+r−

`r² dt 2

12(n³−n)δ_n+m,₀ c= 3` 2G

= 6k

4π Z

M

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IEH3[g] = 1 16πG

Z

M

d³x√

−g R+ 2

`²

+ ˆI∂M

`²r² dt²+ `²r²dr²

(r²−r₊²)(r²−r₋²)+r²

dϕ−r+r−

`r² dt 2

12(n³−n)δ_n+m,₀ c= 3`

2G

= 6k

4π Z

M

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IEH3[g] = 1 16πG

Z

M

d³x√

−g R+ 2

`²

+ ˆI∂M

`²r² dt²+ `²r²dr²

(r²−r₊²)(r²−r₋²)+r²

dϕ−r+r−

`r² dt 2

12(n³−n)δ_n+m,₀ c= 3`

2G = 6k

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Hamiltonian analysis of Chern–Simons theory

I Hamiltonian action of Chern–Simons theory on cylinder adapted coordinates: r: radius,σ ∼σ+ 2π: angle,t: time

ICS[A] = k 4π

Z

M

Tr A_rA˙_σ−A_σA˙_r+ 2A_tF_σr +I∂M

I constraintFσr = 0 locally solved by

Ai=G⁻¹∂iG G∈SL(2,R) I gauge ∂σAr=A⁰_r= 0 impliesG=g(t, σ)b(t, r)

A_σ =b⁻¹a_σb a_σ =g⁻¹g⁰ A_r=b⁻¹∂_rb I for formulating boundary conditions related convenient Ansatz:

A(t, σ, r) =b⁻¹(r) d+a(t, σ)

b(r) a=a_tdt+a_σdσ with vanishing variation δb= 0 and allowed variationsδa6= 0

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ICS[A] = k 4π

Z

M

Ai=G⁻¹∂iG G∈SL(2,R)

I gauge ∂σAr=A⁰_r= 0 impliesG=g(t, σ)b(t, r)

A(t, σ, r) =b⁻¹(r) d+a(t, σ)

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ICS[A] = k 4π

Z

M

Ai=G⁻¹∂iG G∈SL(2,R) I gauge∂σAr=A⁰_r= 0 impliesG=g(t, σ)b(t, r)

A_σ =b⁻¹a_σb a_σ =g⁻¹g⁰ A_r=b⁻¹∂_rb

I for formulating boundary conditions related convenient Ansatz: A(t, σ, r) =b⁻¹(r) d+a(t, σ)

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ICS[A] = k 4π

Z

M

Ai=G⁻¹∂iG G∈SL(2,R) I gauge∂σAr=A⁰_r= 0 impliesG=g(t, σ)b(t, r)

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Holonomies and boundary action

I locally Chern–Simons is trivial, but globally holonomies can exist

I encode holonomies in (non-)periodicity properties of group elementg g(t, σ+2π) =h g(t, σ) h∈SL(2,R) Trh= Tr

Pexp I

a_σdσ

assume for simplicity time-independence ofh

I Hamiltonian action decomposes into three terms ICS[A] =− k

4π Z

∂M

dtdσTr g⁰g⁻¹gg˙ ⁻¹

− k 12π

Z

M

Tr G⁻¹dG3

+I∂M

I Gauss decomposition G=e^XL⁺e^ΦL⁰e^{Y L}⁻ yields boundary action ICS[Φ, X, Y] =− k

4π Z

∂M

dtdσ ¹₂ΦΦ˙ ⁰−2e^ΦX⁰Y˙ +I∂M used standard basis for SL(2,R): [Ln, Lm] = (n−m)Ln+mforn, m= 0,±1

also used Polyakov–Wiegmann identity to showb-independence of action and choseb= 1at∂M

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I locally Chern–Simons is trivial, but globally holonomies can exist I encode holonomies in (non-)periodicity properties of group elementg g(t, σ+2π) =h g(t, σ) h∈SL(2,R) Trh= Tr

Pexp I

a_σdσ

4π Z

∂M

− k 12π

Z

M

Tr G⁻¹dG3

+I∂M

4π Z

∂M

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Pexp I

a_σdσ

4π Z

∂M

− k 12π

Z

M

Tr G⁻¹dG3

+I∂M

4π Z

∂M

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Pexp I

a_σdσ

4π Z

∂M

− k 12π

Z

M

Tr G⁻¹dG3

+I∂M

I Gauss decomposition G=e^XL⁺e^ΦL⁰e^{Y L}⁻ yields boundary action k Z

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Outline

Overture

KdV deformation

Conclusions

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Near horizon boundary conditions (metric formulation) so far have not imposed any boundary conditions

I consider near horizon expansion ds² =−κ²r² dt²+dr²+^`₄² J⁺+J⁻2

dσ²+κ J⁺− J⁻

r² dtdσ+. . . r →0: Rindler horizon

κ: surface gravity

J⁺(t, σ) +J⁻(t, σ): metric transversal to horizon

. . .: terms of higher order inr

I assumption 1: impose boundary conditions on (stretched) horizon, not at infinity

I assumption 2: surface gravity state-independent, δκ= 0

I assumption 3: other metric functions state-dependent, δJ^±6= 0 I simplifying assumption: constantsurface gravity ⇒ “holographic

Ward identities” imply time-independence of state-dependent fct’s J˙^±= 0

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I consider near horizon expansion ds²=−κ²r² dt²+dr²+^`₄² J⁺+J⁻2

r² dtdσ+. . . r→0: Rindler horizon

κ: surface gravity

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κ: surface gravity

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κ: surface gravity

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κ: surface gravity

I assumption 3: other metric functions state-dependent, δJ^±6= 0

I simplifying assumption: constantsurface gravity ⇒ “holographic Ward identities” imply time-independence of state-dependent fct’s

J˙^±= 0

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κ: surface gravity

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Black holes can be deformed into black flowersAfshar et al. 16 Horizon can get excited by area preserving shear-deformations

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Near horizon Chern–Simons connection

I same boundary conditions in Chern–Simons language:

a= J(σ) dσ−κdt

L₀ A=b⁻¹ d+a b

I boundary condition preserving gauge trafos δεa= dε+ [a, ε]: δεJ =η⁰ ε=η L0+. . .

I canonical boundary charges in general: δQ[ε] =− k

2π I

dσTr ε δaσ

I canonical boundary charges for near horizon boundary conditions: Q[η] =− k

4π I

dσ ηJ

I like Brown–Henneaux: 2 towers of conserved boundary chargesJ^±

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a= J(σ) dσ−κdt

L₀ A=b⁻¹ d+a b I boundary condition preserving gauge trafos δεa= dε+ [a, ε]:

δεJ =η⁰ ε=η L0+. . .

I canonical boundary charges in general: δQ[ε] =− k

2π I

dσTr ε δaσ

4π I

dσ ηJ

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a= J(σ) dσ−κdt

δεJ =η⁰ ε=η L0+. . . I canonical boundary charges in general:

δQ[ε] =− k 2π

I

dσTr ε δaσ

4π I

dσ ηJ

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a= J(σ) dσ−κdt

δQ[ε] =− k 2π

I

dσTr ε δaσ

I canonical boundary charges for near horizon boundary conditions:

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a= J(σ) dσ−κdt

δQ[ε] =− k 2π

I

dσTr ε δaσ

I canonical boundary charges for near horizon boundary conditions:

Q[η] =− k 4π

I

dσ ηJ

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Near horizon symmetries

I near horizon symmetries = all boundary condition preserving trafos modulo trivial gauge trafos

I near horizon symmetries generated by canonical boundary charges δη1Q[η2] ={Q[η₁], Q[η2]}=− k

4π I

dσ η2η₁⁰ I introduce Fourier modes

Jn= 1 2π

I

dσJ e^inσ

I find two affineu(1)current algebras as near horizon symmetries [Jn, Jm] = 2

kn δn+m,0

replaced Poisson brackets by commutators as usual,i{,} →[,]; note: algebra isomorphic toHeisenbergalgebras

I simpler than Brown–Henneaux, who found Virasoros

the Brown–Henneaux Virasoros are recovered unambiguously through a twisted Sugawara-construction

I near-horizon (Cardy-like) entropy formula: S= 2π J₀⁺+J₀⁻

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4π I

dσ η2η₁⁰

I introduce Fourier modes

Jn= 1 2π

I

dσJ e^inσ

kn δn+m,0

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4π I

Jn= 1 2π

I

dσJ e^inσ

kn δn+m,0

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4π I

Jn= 1 2π

I

dσJ e^inσ

kn δn+m,0

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4π I

Jn= 1 2π

I

dσJ e^inσ

kn δn+m,0

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4π I

Jn= 1 2π

I

dσJ e^inσ

kn δn+m,0

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Unique features of near horizon boundary conditions 1. All states allowed by bc’s have same temperature

By contrast: asymptotically AdS or flat space bc’s allow for black hole states at different masses and hence different temperatures

2. All states allowed by bc’s are regular

(in particular, they have no conical singularities at the horizon in the Euclidean formulation)

3. There is a non-trivial reducibility parameter (= Killing vector) 4. Technical feature: in Chern–Simons formulation of 3d gravity simple

expressions in diagonal gauge

A^±=b^∓1 d+a^± b^±1 a^±=L0 J^± dσ−κ dt

b= exp

L₊−L− r/2 near horizon metric recovered from

gµν = `²

2 Tr (A⁺_µ −A⁻_µ)(A⁺_ν −A⁻_ν) 5. Leads to soft Heisenberg hair (see next slide!)

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Unique features of near horizon boundary conditions 1. All states allowed by bc’s have same temperature 2. All states allowed by bc’s are regular

By contrast: for given temperature not all states in theories with asymptotically AdS or flat space bc’s are free from conical singularities; usually a unique black hole state is picked

A^±=b^∓1 d+a^± b^±1 a^±=L0 J^± dσ−κ dt

b= exp

L+−L−

r/2 near horizon metric recovered from

g_µν = `²

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3. There is a non-trivial reducibility parameter (= Killing vector)

By contrast: for any other known (non-trivial) bc’s there is no vector field that is Killing for all geometries allowed by bc’s

4. Technical feature: in Chern–Simons formulation of 3d gravity simple expressions in diagonal gauge

A^±=b^∓1 d+a^± b^±1 a^±=L0 J^± dσ−κ dt

b= exp

gµν = `²

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A^±=b^∓1 d+a^± b^±1 a^±=L₀ J^± dσ−κ dt

b= exp

gµν = `²

2 Tr (A⁺_µ −A⁻_µ)(A⁺_ν −A⁻_ν)

5. Leads to soft Heisenberg hair (see next slide!)

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A^±=b^∓1 d+a^± b^±1 a^±=L₀ J^± dσ−κ dt

b= exp

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Soft Heisenberg hair

I Black flower excitations = hair of black holes Algebraically, excitations from descendants

|black floweri ∼ Y

n^±_i>0

J⁺

−n⁺_i J⁻

−n⁻_i |black holei

I What is energy of such excitations?

I Near horizon Hamiltonian = boundary charge associated with unit time-translations

H =Q[∂t] =κ J₀⁺+J₀⁻ commutes with all generators J_n^±

I H-eigenvalue of black flower =H-eigenvalue of black hole I Black flower excitations do not change energy of black hole!

Black flower excitations = soft hair in sense of Hawking, Perry and Strominger ’16

Call it “soft Heisenberg hair”

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n^±_i>0

J⁺

−n⁺_i J⁻

−n⁻_i |black holei I What is energy of such excitations?

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n^±_i>0

J⁺

−n⁺_i J⁻

I Near horizon Hamiltonian = boundary charge associated with unit time-translations^∗

∗ units defined by specifyingκ

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n^±_i>0

J⁺

−n⁺_i J⁻

I H-eigenvalue of black flower =H-eigenvalue of black hole

I Black flower excitations do not change energy of black hole! Black flower excitations = soft hair in sense of Hawking, Perry and Strominger ’16

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n^±_i>0

J⁺

−n⁺_i J⁻

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n^±_i>0

J⁺

−n⁺_i J⁻

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n^±_i>0

J⁺

−n⁺_i J⁻

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Outline

Overture

KdV deformation

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Near horizon boundary action I recall general boundary action

ICS[Φ, X, Y] =− k 4π

Z

∂M

dtdσ ¹₂ΦΦ˙ ⁰−2e^ΦX⁰Y˙

+I∂M

I near horizon boundary conditions imply

Φ⁰ =J X⁰= 0 I scalar field Φhas generalized periodicity property

Φ(t, σ+ 2π) = Φ(t, σ) + 2πJ₀

I near horizon boundary action simplifies ICS[Φ] =− k

4π Z

∂M

dtdσ¹₂ΦΦ˙ ⁰+I∂M

I still need to discussI∂M, since it encodes the boundary Hamiltonian!

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Z

∂M

+I∂M

Φ⁰ =J X⁰= 0

I scalar field Φhas generalized periodicity property Φ(t, σ+ 2π) = Φ(t, σ) + 2πJ₀

4π Z

∂M

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Z

∂M

+I∂M

Φ(t, σ+ 2π) = Φ(t, σ) + 2πJ₀

4π Z

∂M

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Z

∂M

+I∂M

Φ(t, σ+ 2π) = Φ(t, σ) + 2πJ₀ I near horizon boundary action simplifies

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Z

∂M

+I∂M

Φ(t, σ+ 2π) = Φ(t, σ) + 2πJ₀

4π Z

∂M

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Simplest choice of boundary term I well-defined variational principle if

δI_∂M= k 2π

Z

∂M

dtdσTr a_tδa_σ

I defininga_t=−ζ(t, σ)L₀ and using near horizon boundary conditions for aσ yields

δI∂M = k 2π

Z

∂M

dtdσζδJ I integrability of boundary action requires

ζ(J) = δH δJ whereH is the boundary Hamiltonian density

I simplest choice (near horizon boundary conditions for a_t): δζ = 0

make this choice to obtain near horizon Hamiltonian!

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δI_∂M= k 2π

Z

∂M

dtdσTr a_tδa_σ

δI∂M = k 2π

Z

∂M

dtdσζδJ

I integrability of boundary action requires ζ(J) = δH

δJ whereH is the boundary Hamiltonian density

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δI_∂M= k 2π

Z

∂M

dtdσTr a_tδa_σ

δI∂M = k 2π

Z

∂M

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δI_∂M= k 2π

Z

∂M

dtdσTr a_tδa_σ

δI∂M = k 2π

Z

∂M

I simplest choice (near horizon boundary conditions fora_t):

δζ = 0

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I solving integrability condition

ζ(J) = δH δJ for Hyields boundary Hamiltonian density

H_NH=ζJ =ζΦ⁰

I this was the main result announced in the beginning I full boundary action given by

INH[Φ] =− k 2π

Z

dtdσ1 2

ΦΦ˙ ⁰+ζΦ⁰

⇒ momentum given by spatial derivative,Π∼Φ⁰! I near horizon Hamiltonian given by zero mode generator

HNH = k 2π

I

dσHNH= k 2ζJ0

recovers result expected from near horizon symmetry analysis

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H_NH=ζJ =ζΦ⁰

I this was the main result announced in the beginning

I full boundary action given by INH[Φ] =− k 2π

Z

dtdσ1 2

ΦΦ˙ ⁰+ζΦ⁰

HNH = k 2π

I

dσHNH= k 2ζJ0

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H_NH=ζJ =ζΦ⁰

INH[Φ] =− k 2π

Z

dtdσ1 2

ΦΦ˙ ⁰+ζΦ⁰

⇒ momentum given by spatial derivative,Π∼Φ⁰!

I near horizon Hamiltonian given by zero mode generator HNH = k

2π I

dσHNH= k 2ζJ0

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H_NH=ζJ =ζΦ⁰

INH[Φ] =− k 2π

Z

dtdσ1 2

ΦΦ˙ ⁰+ζΦ⁰

HNH = k 2π

I

dσHNH= k 2ζJ0

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Mode decomposition

I near horizon equations of motion Φ˙⁰ = 0 solved by

Φ(t, σ)

EOM= Φ₀(t) +J₀σ+X

n6=0

J_n in e^inσ

I off-shell similar mode-decomposition

Φ(t, σ) = Φ₀(t) +J₀(t)σ+X

n6=0

J_n(t) in e^inσ due to generalized periodicty property ofΦ

I time-independence of holonomy requires J˙0 = 0

I off-shell mode-decomposition in near horizon boundary action: INH[Φ0,Jn] = k

2 Z

dt

− 1

2Φ˙0J0+X

n>0

i

nJ˙nJ−n−ζJ0

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Mode decomposition

Φ(t, σ)

n6=0

J_n in e^inσ I off-shell similar mode-decomposition

Φ(t, σ) = Φ₀(t) +J₀(t)σ+X

n6=0

2 Z

dt

− 1

2Φ˙0J0+X

n>0

i

nJ˙nJ−n−ζJ0

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Mode decomposition

Φ(t, σ)

n6=0

J_n in e^inσ I off-shell similar mode-decomposition

Φ(t, σ) = Φ₀(t) +J₀(t)σ+X

n6=0

2 Z

dt

− 1

2Φ˙0J0+X

n>0

i

nJ˙nJ−n−ζJ0