Soft Heisenberg Hair - quark.itp.tuwien.ac.atquark.itp.tuwien.ac.at/~grumil/pdf/nordita2017.pdf · Soft Heisenberg Hair Daniel Grumiller Institute for Theoretical Physics TU Wien

Soft Heisenberg Hair

Daniel Grumiller

Institute for Theoretical PhysicsTU Wien

Nordita + Stockholm University, April 2017

1603.04824, 1607.00009, 1607.05360, 1611.09783, 1703.02594

https://arxiv.org/abs/1603.04824





Daniel Grumiller — Soft Heisenberg Hair 2/32

Two simple punchlines

1. Heisenberg algebra[Xn, Pm] = i δn,m

fundamental not only in quantum mechanicsbut also in near horizon physics of (higher spin) gravity theories

2. Black hole microstates identified as specific “soft hair” descendants

based on work withI Hamid Afshar [IPM Teheran]I Stephane Detournay [ULB]I Wout Merbis [TU Wien]I Blagoje Oblak [ULB / ETH]I Alfredo Perez [CECS Valdivia]I Stefan Prohazka [TU Wien]I Shahin Sheikh-Jabbari [IPM Teheran]I David Tempo [CECS Valdivia]I Ricardo Troncoso [CECS Valdivia]














Outline

Motivation

Near horizon boundary conditions

Explicit construction of BTZ microstates

Discussion


Outline

Motivation



Discussion

Daniel Grumiller — Soft Heisenberg Hair Motivation 5/32

Black hole microstates

Bekenstein–Hawking

SBH =A

4GNI Motivation: microscopic understanding of generic black hole entropy

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

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

I Main idea: consider near horizon symmetries for non-extremalhorizons

I Near horizon line-element with Rindler acceleration a:

ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .

Meaning of coordinates:I ρ: radial direction (ρ = 0 is horizon)I ϕ ∼ ϕ+ 2π: angular directionI v: (advanced) time




SBH =A

4GNI Motivation: microscopic understanding of generic black hole entropyI Microstate counting from Cardyology (Strominger, Carlip, ...)

using CFT2 symmetries and Cardy formula




ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .





SBH =A




warped CFT: Detournay, Hartman, Hofman ’12Galilean CFT: Bagchi, Detournay, Fareghbal, Simon ’13; Barnich ’13



ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .





SBH =A





Related ideas pursued e.g. byI Donnay, Giribet, Gonzalez, Pino ’15I Hawking, Perry, Strominger ’16I ...

Postpone comparison with related approaches after discussing our approach


ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .





SBH =A






ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .



Choices

I Rindler acceleration: state-dependent or chemical potential?

I If state-dependent: need mechanism to fix scale

— suggestion in1512.08233:

v ∼ v + 2πL

Works technically but physical interpretation difficult

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

TU =a

2π

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

ICS = ±∑±

k

4π

∫〈A± ∧ dA± + 2

3A± ∧A± ∧A±〉

with sl(2) connections A± and k = `/(4GN ) with AdS radius ` = 1


http://arxiv.org/abs/1512.08233

Choices

I Rindler acceleration: state-dependent or chemical potential?I If state-dependent: need mechanism to fix scale

— suggestion in1512.08233:

v ∼ v + 2πL


Recall scale invariance

a→ λa ρ→ λρ v→v/λof Rindler metric

ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2


TU =a

2πI Work in 3d Einstein gravity in Chern–Simons formulation

ICS = ±∑±

k

4π

∫〈A± ∧ dA± + 2

3A± ∧A± ∧A±〉




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


Recall scale invariance

a→ λa ρ→ λρ v→v/λof Rindler metric

ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2


TU =a

2πI Work in 3d Einstein gravity in Chern–Simons formulation

ICS = ±∑±

k

4π

∫〈A± ∧ dA± + 2

3A± ∧A± ∧A±〉




Choices


I If state-dependent: need mechanism to fix scale — suggestion in1512.08233:

v ∼ v + 2πL


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

TU =a

2π


ICS = ±∑±

k

4π

∫〈A± ∧ dA± + 2

3A± ∧A± ∧A±〉




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 difficultI If chemical potential: all states in theory have same

(Unruh-)temperature

TU =a

2πsuggestion in 1511.08687

We make this choice in this talk!


ICS = ±∑±

k

4π

∫〈A± ∧ dA± + 2

3A± ∧A± ∧A±〉





Choices


I If state-dependent: need mechanism to fix scale — suggestion in1512.08233:

v ∼ v + 2πL


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

TU =a

2π


ICS = ±∑±

k

4π

∫〈A± ∧ dA± + 2

3A± ∧A± ∧A±〉




Outline

Motivation



Discussion

Daniel Grumiller — Soft Heisenberg Hair Near horizon boundary conditions 8/32

Review of general aspects of boundary conditions (bc’s)

I In any physical theory need bc’s imposed on fields

I In many instances ‘natural’ bc’s suitableI In gravity ‘natural’ bc’s most unnatural: metric cannot be assumed to

vanish asymptoticallyI Instead, metric should approach some suitable class of metrics, like

asymptotically flat or asymptotically (A)dSI No algorithm determining ‘right’ bc’s — always choice!I Algorithm exists to check consistency of bc’sI Local diffeos and gauge trafos fall into three classes:

1. Trafos that violate bc’s (forbidden)2. Trafos that preserve bc’s and remain pure gauge (trivial)3. Trafos that preserve bc’s but are not pure gauge at the asymptotic

boundary (asymptotic symmetries)I Canonical boundary charges (a la Regge–Teitelboim) generate

asympotic symmetries of “edge states”I Consistency means they are finite, integrable, non-trivial and

conserved (in time)



I In any physical theory need bc’s imposed on fieldsI In many instances ‘natural’ bc’s suitable

Example:Φ(x→∞) = 0

I In gravity ‘natural’ bc’s most unnatural: metric cannot be assumed tovanish asymptotically

I Instead, metric should approach some suitable class of metrics, likeasymptotically flat or asymptotically (A)dS

I No algorithm determining ‘right’ bc’s — always choice!I Algorithm exists to check consistency of bc’sI Local diffeos and gauge trafos fall into three classes:




conserved (in time)



I In any physical theory need bc’s imposed on fieldsI In many instances ‘natural’ bc’s suitableI In gravity ‘natural’ bc’s most unnatural: metric cannot be assumed to

vanish asymptotically

I Instead, metric should approach some suitable class of metrics, likeasymptotically flat or asymptotically (A)dS





conserved (in time)





asymptotically flat or asymptotically (A)dS

Example: Brown-Henneaux type of bc’s (aAdS3):

ds2aAdS = dρ2 +

(e2ρηµν + γµν +O(e−2ρ)

)dxµ dxν

with δγ = arbitrary





conserved (in time)





asymptotically flat or asymptotically (A)dSI No algorithm determining ‘right’ bc’s — always choice!

I Algorithm exists to check consistency of bc’sI Local diffeos and gauge trafos fall into three classes:




conserved (in time)





asymptotically flat or asymptotically (A)dSI No algorithm determining ‘right’ bc’s — always choice!I Algorithm exists to check consistency of bc’s

I Local diffeos and gauge trafos fall into three classes:1. Trafos that violate bc’s (forbidden)2. Trafos that preserve bc’s and remain pure gauge (trivial)3. Trafos that preserve bc’s but are not pure gauge at the asymptotic



conserved (in time)







boundary (asymptotic symmetries)

I Canonical boundary charges (a la Regge–Teitelboim) generateasympotic symmetries of “edge states”

I Consistency means they are finite, integrable, non-trivial andconserved (in time)








asympotic symmetries of “edge states”

I Consistency means they are finite, integrable, non-trivial andconserved (in time)









conserved (in time)


AdS3 bc’s

Even restricting to Einstein gravity in three dimensions (with negativecosmological constant) different choices exist for bc’s and their associatedasymptotic symmetry algebras:

I Brown–Henneaux (1986): two Virasoros (2d conformal algebra)

I Compere–Song–Strominger (2013): Virasoro plus u(1) current algebra

I Troessaert (2013): 2 Virasoros plus 2 u(1) current algebras

I Avery–Poojary–Suryanarayana (2013): Virasoro plus sl(2) currentalgebra

I Donnay–Giribet–Gonzalez–Pino (2015): centerless warped conformal

I Afshar–Detournay–DG–Oblak (2015): twisted warped conformal

I DG–Riegler (2016): two sl(2) current algebras (most general case!)

Our near horizon bc’s simpler than any of the above!


AdS3 bc’s











AdS3 bc’s











AdS3 bc’s











Explicit specification of our bc’s in 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 andboundary 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, 1I 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 dϕL0

and b = exp (1ζ L+) · exp (ρ2 L−). (assume constant ζ for simplicity)




A = b−1(ρ)(

d+a(x0, x1))b(ρ)



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







A = b−1(ρ)(

d+a(x0, x1))b(ρ)



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





Near horizon metric

Usinggµν = 1

2

⟨(A+µ −A−µ

) (A+ν −A−ν

)⟩

yields (f := 1 + ρ/(2a))

ds2 = −2aρf dv2 + 2 dv dρ− 2ωa−1 dϕdρ

+ 4ωρf dv dϕ+[γ2 + 2ρ

a f(γ2 − ω2)]

dϕ2

state-dependent functions J ± = γ ± ω, chemical potentials ζ± = −a± ΩNeglecting rotation terms (ω = 0) yields Rindler plus higher order terms:

ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .

Comments:

I Recover desired near horizon metric

I Rindler acceleration a indeed state-independentI Two state-dependent functions (γ, ω) as “usual” in 3d gravityI γ = γ(ϕ): “black flower”


Near horizon metric

Usinggµν = 1

2

⟨(A+µ −A−µ

) (A+ν −A−ν

)⟩yields (f := 1 + ρ/(2a))



a f(γ2 − ω2)]

dϕ2

state-dependent functions J ± = γ ± ω, chemical potentials ζ± = −a± Ω

For simplicity set Ω = 0 and a = const. in metric above

EOM imply ∂vJ ± = ±∂ϕζ±; in this case ∂vJ ± = 0

Neglecting rotationterms (ω = 0) yields Rindler plus higher order terms:

ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .

Comments:I Recover desired near horizon metric



Near horizon metric

Usinggµν = 1

2

⟨(A+µ −A−µ

) (A+ν −A−ν

)⟩yields (f := 1 + ρ/(2a))



a f(γ2 − ω2)]

dϕ2


ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .

Comments:

I Recover desired near horizon metric



Near horizon metric

Usinggµν = 1

2

⟨(A+µ −A−µ

) (A+ν −A−ν

)⟩yields (f := 1 + ρ/(2a))



a f(γ2 − ω2)]

dϕ2


ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .

Comments:

I Recover desired near horizon metricI Rindler acceleration a indeed state-independent

I Two state-dependent functions (γ, ω) as “usual” in 3d gravityI γ = γ(ϕ): “black flower”


Near horizon metric

Usinggµν = 1

2

⟨(A+µ −A−µ

) (A+ν −A−ν

)⟩yields (f := 1 + ρ/(2a))



a f(γ2 − ω2)]

dϕ2


ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .

Comments:

I Recover desired near horizon metricI Rindler acceleration a indeed state-independentI Two state-dependent functions (γ, ω) as “usual” in 3d gravity

I γ = γ(ϕ): “black flower”


Near horizon metric

Usinggµν = 1

2

⟨(A+µ −A−µ

) (A+ν −A−ν

)⟩yields (f := 1 + ρ/(2a))



a f(γ2 − ω2)]

dϕ2


ds2 = −2aρ dv2 + 2 dv dρ+ γ2 dϕ2 + . . .

Comments:

I Recover desired near horizon metricI Rindler acceleration a indeed state-independentI Two state-dependent functions (γ, ω) as “usual” in 3d gravityI γ = γ(ϕ): “black flower”


Canonical boundary charges

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

I Zero mode charges: mass and angular momentum

Background independent result for Chern–Simons yields

Q[η] =k

4π

∮dϕη(ϕ)J (ϕ)

I FiniteI IntegrableI ConservedI Non-trivial

Meaningful near horizon boundaryconditions and non-trivial theory!






Q[η] =k

4π

∮dϕη(ϕ)J (ϕ)








Q[η] =k

4π

∮dϕη(ϕ)J (ϕ)




Near horizon symmetry algebra

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

Most general trafo

δεa = dε+ [a, ε] = O(δa)

that preserves our boundary conditions for constant ζ given by

ε = ε+L+ + ηL0 + ε−L−

with∂vη = 0

implyingδεJ = ∂ϕη

I Expand charges in Fourier modes

J±n =k

4π

∮dϕeinϕJ ± (ϕ)

I Near horizon symmetry algebra[J±n , J

±m

]= ±1

2knδn+m, 0

[J+n , J

−m

]= 0

Two u(1) current algebras with non-zero levelsI Much simpler than CFT2, warped CFT2, Galilean CFT2, etc.I Map

P0 = J+0 + J−0 Pn = i

kn (J+−n + J−−n) if n 6= 0 Xn = J+

n − J−nyields Heisenberg algebra (with Casimirs X0, P0)

[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0

[Xn, Pm]= iδn,m if n 6= 0





J±n =k

4π


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 theirnear horizon analysis)


±m

]= ±1

2knδn+m, 0

[J+n , J

−m

]= 0


P0 = J+0 + J−0 Pn = i

kn (J+−n + J−−n) if n 6= 0 Xn = J+


[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0






J±n =k

4π



±m

]= ±1

2knδn+m, 0

[J+n , J

−m

]= 0

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

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

P0 = J+0 + J−0 Pn = i

kn (J+−n + J−−n) if n 6= 0 Xn = J+


[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0






J±n =k

4π



±m

]= ±1

2knδn+m, 0

[J+n , J

−m

]= 0

Two u(1) current algebras with non-zero levelsI Much simpler than CFT2, warped CFT2, Galilean CFT2, etc.

I Map

P0 = J+0 + J−0 Pn = i

kn (J+−n + J−−n) if n 6= 0 Xn = J+


[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0






J±n =k

4π



±m

]= ±1

2knδn+m, 0

[J+n , J

−m

]= 0


P0 = J+0 + J−0 Pn = i

kn (J+−n + J−−n) if n 6= 0 Xn = J+


[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0



Macroscopic entropy

Using any of the usual methods to determine entropy yields

S = 2π(J+

0 + J−0)

Properties of result:

I Simple (compare with Cardy-type of formulas!)

I Fairly universal (AdS, flat space, higher spins, ...)

e.g. entropy for BTZ black holes in spin-3 gravity:asymptotic result

S = 2π√

2πk

(√L+ cos

[1

3arcsin

(3

8

√3k

2πL3+

W+

)]

+√L− cos

[1

3arcsin

(3

8

√3k

2πL3−W−

)])

equivalent to simpler near horizon result above!


Macroscopic entropy


S = 2π(J+

0 + J−0)





S = 2π√

2πk

(√L+ cos

[1

3arcsin

(3

8

√3k

2πL3+

W+

)]

+√L− cos

[1

3arcsin

(3

8

√3k

2πL3−W−

)])



Macroscopic entropy


S = 2π(J+

0 + J−0)





S = 2π√

2πk

(√L+ cos

[1

3arcsin

(3

8

√3k

2πL3+

W+

)]

+√L− cos

[1

3arcsin

(3

8

√3k

2πL3−W−

)])



Macroscopic entropy


S = 2π(J+

0 + J−0)





S = 2π√

2πk

(√L+ cos

[1

3arcsin

(3

8

√3k

2πL3+

W+

)]

+√L− cos

[1

3arcsin

(3

8

√3k

2πL3−W−

)])

equivalent to simpler near horizon result above!Daniel Grumiller — Soft Heisenberg Hair Near horizon boundary conditions 15/32

Brief list of generalizations

Heisenberg algebras as near horizon symmetries arise not only in AdS3

Einstein gravity, but also in ...

I ... flat space Einstein gravity in three dimensionsAfshar, DG, Merbis, Perez, Tempo, Troncoso ’16

I ... higher spin gravity in three dimensionsDG, Perez, Prohazka, Tempo, Troncoso ’16

I ... higher derivative gravity in three dimensionsSetare, Adami ’16

I ... general relativity (in four dimensions)Afshar, DG, Sheikh-Jabbari ’16

I ... flat space higher spin gravity in three dimensionsAmmon, Grumiller, Prohazka, Riegler, Wutte ’17

Conclusions about near horizon symmetry algebra fairly general!


Outline

Motivation



Discussion

Daniel Grumiller — Soft Heisenberg Hair Explicit construction of BTZ microstates 17/32

Near horizon Hilbert space

I Denote “near horizon” generators with calligraphic letters

I Near horizon algebra (conveniently rescaled)

[J ±n , J ±m ] = 12 n δn,−m

I Near horizon Hilbert space: define vacuum by highest weightconditions

J ±n |0〉 = 0 for all n ≥ 0.

I Construct near horizon Virasoro through standard Sugawaraconstruction

L±n ≡∑p∈Z

:J ±n−p J ±p :

I Get Virasoro algebra with central charge 1

[L±n ,L±m] = (n−m)L±n+m + 112(n3 − n)δn,−m

[L±n ,J ±m ] = −mJ ±n+m

I Call this “near horizon symmetry algebra” (note: independent from `)



I Denote “near horizon” generators with calligraphic lettersI Near horizon algebra (conveniently rescaled)

[J ±n , J ±m ] = 12 n δn,−m


J ±n |0〉 = 0 for all n ≥ 0.


L±n ≡∑p∈Z

:J ±n−p J ±p :


[L±n ,L±m] = (n−m)L±n+m + 112(n3 − n)δn,−m

[L±n ,J ±m ] = −mJ ±n+m





[J ±n , J ±m ] = 12 n δn,−m


J ±n |0〉 = 0 for all n ≥ 0.


L±n ≡∑p∈Z

:J ±n−p J ±p :


[L±n ,L±m] = (n−m)L±n+m + 112(n3 − n)δn,−m

[L±n ,J ±m ] = −mJ ±n+m





[J ±n , J ±m ] = 12 n δn,−m


J ±n |0〉 = 0 for all n ≥ 0.


L±n ≡∑p∈Z

:J ±n−p J ±p :


[L±n ,L±m] = (n−m)L±n+m + 112(n3 − n)δn,−m

[L±n ,J ±m ] = −mJ ±n+m





[J ±n , J ±m ] = 12 n δn,−m


J ±n |0〉 = 0 for all n ≥ 0.


L±n ≡∑p∈Z

:J ±n−p J ±p :


[L±n ,L±m] = (n−m)L±n+m + 112(n3 − n)δn,−m

[L±n ,J ±m ] = −mJ ±n+m





[J ±n , J ±m ] = 12 n δn,−m


J ±n |0〉 = 0 for all n ≥ 0.


L±n ≡∑p∈Z

:J ±n−p J ±p :


[L±n ,L±m] = (n−m)L±n+m + 112(n3 − n)δn,−m

[L±n ,J ±m ] = −mJ ±n+m



Soft hair

I Generic descendant of vacuum:

|Ψ(n±i )〉 =∏n±i >0

(J +

−n+i

J −−n−i)|0〉

with set of positive integers n±i > 0

I Near horizon Hamiltonian H ∼ J +0 + J −0 commutes with near

horizon symmetry algebra

I Descendants of vacuum have zero energy; dubbed “soft hair” (sametrue for descendants of black holes)

I Immediate issue for entropy: infinite soft hair degeneracy!

I Note: descendants have positive eigenvalues of L±0

L±0 |Ψ(n±i )〉 =∑i

n±i |Ψ(n±i )〉 ≡ E±Ψ |Ψ(n±i )〉

I Will exploit this property to provide cut-off on soft hair spectrum!


Soft hair


|Ψ(n±i )〉 =∏n±i >0

(J +

−n+i

J −−n−i)|0〉

with set of positive integers n±i > 0I Near horizon Hamiltonian H ∼ J +

0 + J −0 commutes with nearhorizon symmetry algebra




L±0 |Ψ(n±i )〉 =∑i

n±i |Ψ(n±i )〉 ≡ E±Ψ |Ψ(n±i )〉



Soft hair


|Ψ(n±i )〉 =∏n±i >0

(J +

−n+i

J −−n−i)|0〉






L±0 |Ψ(n±i )〉 =∑i

n±i |Ψ(n±i )〉 ≡ E±Ψ |Ψ(n±i )〉



Soft hair


|Ψ(n±i )〉 =∏n±i >0

(J +

−n+i

J −−n−i)|0〉






L±0 |Ψ(n±i )〉 =∑i

n±i |Ψ(n±i )〉 ≡ E±Ψ |Ψ(n±i )〉



Soft hair


|Ψ(n±i )〉 =∏n±i >0

(J +

−n+i

J −−n−i)|0〉






L±0 |Ψ(n±i )〉 =∑i

n±i |Ψ(n±i )〉 ≡ E±Ψ |Ψ(n±i )〉



Soft hair


|Ψ(n±i )〉 =∏n±i >0

(J +

−n+i

J −−n−i)|0〉






L±0 |Ψ(n±i )〉 =∑i

n±i |Ψ(n±i )〉 ≡ E±Ψ |Ψ(n±i )〉



Asymptotic Hilbert space

I Our algebra in original form

[J±n , J±m] = k

2 n δn,−m

I Map to asymptotic Virasoro algebra through twisted Sugawaraconstruction

[L±n , J±m] = −mJ±n+m + ik2 m

2 δn,−m

Note to experts: twist-term follows uniquely from mapping ourconnection into highest weight gauge

I Get asymptotic Virasoro with Brown–Henneaux central charge

[L±n , L±m] = (n−m)L±n+m + c

12 n3 δn,−m

where c = 6k = 3`/(2GN ) in large k-limitI Algebra gets twisted

[L±n , J±m] = −mJ±n+m + ik2 m

2 δn,−m




[J±n , J±m] = k

2 n δn,−m


[L±n , J±m] = −mJ±n+m + ik2 m

2 δn,−m



[L±n , L±m] = (n−m)L±n+m + c

12 n3 δn,−m


[L±n , J±m] = −mJ±n+m + ik2 m

2 δn,−m




[J±n , J±m] = k

2 n δn,−m


[L±n , J±m] = −mJ±n+m + ik2 m

2 δn,−m



[L±n , L±m] = (n−m)L±n+m + c

12 n3 δn,−m

where c = 6k = 3`/(2GN ) in large k-limit

I Algebra gets twisted

[L±n , J±m] = −mJ±n+m + ik2 m

2 δn,−m




[J±n , J±m] = k

2 n δn,−m


[L±n , J±m] = −mJ±n+m + ik2 m

2 δn,−m



[L±n , L±m] = (n−m)L±n+m + c

12 n3 δn,−m


[L±n , J±m] = −mJ±n+m + ik2 m

2 δn,−m


Proposed map between near horizon and asymptotic generators

I Suggestive proposal (see Banados 9811162)

cL±0 = L±0 − 124

I Consistency of algebras then implies

J±n =1√6J ±nc , n 6= 0

I Even though Jn and Jn algebras isomorphic, near horizon algebra hasmore generators than asymptotic one due to relation above, namelyJm with m 6= nc

I Relation between J0 and J0 also induced by above, but not neededI Main point: use eigenvalues of asymptotic generators L±0 (essentially

mass and angular momentum of BTZ) to provide cut-off on soft hairspectrum

〈L±0 〉BTZ = ∆± 〈L±n6=0〉BTZ = 0

I Microstates = all states in near horizon Hilbert space obeyingequations above




cL±0 = L±0 − 124


J±n =1√6J ±nc , n 6= 0




〈L±0 〉BTZ = ∆± 〈L±n6=0〉BTZ = 0





cL±0 = L±0 − 124


J±n =1√6J ±nc , n 6= 0




〈L±0 〉BTZ = ∆± 〈L±n6=0〉BTZ = 0





cL±0 = L±0 − 124


J±n =1√6J ±nc , n 6= 0


I Relation between J0 and J0 also induced by above, but not needed

I Main point: use eigenvalues of asymptotic generators L±0 (essentiallymass and angular momentum of BTZ) to provide cut-off on soft hairspectrum

〈L±0 〉BTZ = ∆± 〈L±n6=0〉BTZ = 0





cL±0 = L±0 − 124


J±n =1√6J ±nc , n 6= 0




〈L±0 〉BTZ = ∆± 〈L±n6=0〉BTZ = 0





cL±0 = L±0 − 124


J±n =1√6J ±nc , n 6= 0




〈L±0 〉BTZ = ∆± 〈L±n6=0〉BTZ = 0



Horizon fluff as microstatesWe are now ready to identify all BTZ microstates

I Vector space VB of BTZ microstates defined by

〈B′|L±n6=0|B〉 = 0 ∀B,B′ ∈ VB

I All BTZ microstates (“horizon fluff”) are then of the form

|B(n±i )〉 = Nn±i ∏0<n±i

(J +

−n+i

J −−n−i)|0〉

subject to 〈L±0 〉BTZ = ∆±

I This is the key resultI Normalization constant Nn±i fixed by compatibility:

〈B′|L±0 |B〉 = ∆±δB,B′

I Useful observation:

∆± = 〈B|L±0 |B〉 ≈1

c〈B|L±0 |B〉 =

1

c

∑i

n±i =1

cE±B




〈B′|L±n6=0|B〉 = 0 ∀B,B′ ∈ VBI All BTZ microstates (“horizon fluff”) are then of the form

|B(n±i )〉 = Nn±i ∏0<n±i

(J +

−n+i

J −−n−i)|0〉



〈B′|L±0 |B〉 = ∆±δB,B′


∆± = 〈B|L±0 |B〉 ≈1

c〈B|L±0 |B〉 =

1

c

∑i

n±i =1

cE±B





|B(n±i )〉 = Nn±i ∏0<n±i

(J +

−n+i

J −−n−i)|0〉


I This is the key result

I Normalization constant Nn±i fixed by compatibility:

〈B′|L±0 |B〉 = ∆±δB,B′


∆± = 〈B|L±0 |B〉 ≈1

c〈B|L±0 |B〉 =

1

c

∑i

n±i =1

cE±B





|B(n±i )〉 = Nn±i ∏0<n±i

(J +

−n+i

J −−n−i)|0〉



〈B′|L±0 |B〉 = ∆±δB,B′


∆± = 〈B|L±0 |B〉 ≈1

c〈B|L±0 |B〉 =

1

c

∑i

n±i =1

cE±B





|B(n±i )〉 = Nn±i ∏0<n±i

(J +

−n+i

J −−n−i)|0〉



〈B′|L±0 |B〉 = ∆±δB,B′


∆± = 〈B|L±0 |B〉 ≈1

c〈B|L±0 |B〉 =

1

c

∑i

n±i =1

cE±B


Microstate counting

I Problem of counting microstates now reduced to combinatorics

I Count all horizon fluff with same energy E±BI Mathematically, reduces to number p(N) of ways positive integer N

partitioned into positive integers

I Work semi-classically, i.e., in limit of large N

I Problem above solved long ago by Hardy and Ramanujan

p(N)∣∣N1

' 1

4N√

3exp

(2π

√N

6

)I Number of BTZ microstates: p(E+

B ) · p(E−B )

I Result for entropy:

S = ln p(E+B ) + ln p(E−B ) = 2π

(√c∆+

6+

√c∆−

6

)+ . . .

I Agrees with Bekenstein–Hawking and Cardy formula


Microstate counting


I Count all horizon fluff with same energy E±B

I Mathematically, reduces to number p(N) of ways positive integer Npartitioned into positive integers



p(N)∣∣N1

' 1

4N√

3exp

(2π

√N

6


B ) · p(E−B )


S = ln p(E+B ) + ln p(E−B ) = 2π

(√c∆+

6+

√c∆−

6

)+ . . .



Microstate counting






p(N)∣∣N1

' 1

4N√

3exp

(2π

√N

6


B ) · p(E−B )


S = ln p(E+B ) + ln p(E−B ) = 2π

(√c∆+

6+

√c∆−

6

)+ . . .



Microstate counting






p(N)∣∣N1

' 1

4N√

3exp

(2π

√N

6


B ) · p(E−B )


S = ln p(E+B ) + ln p(E−B ) = 2π

(√c∆+

6+

√c∆−

6

)+ . . .



Microstate counting






p(N)∣∣N1

' 1

4N√

3exp

(2π

√N

6

)

I Number of BTZ microstates: p(E+B ) · p(E−B )


S = ln p(E+B ) + ln p(E−B ) = 2π

(√c∆+

6+

√c∆−

6

)+ . . .



Microstate counting






p(N)∣∣N1

' 1

4N√

3exp

(2π

√N

6


B ) · p(E−B )


S = ln p(E+B ) + ln p(E−B ) = 2π

(√c∆+

6+

√c∆−

6

)+ . . .



Microstate counting






p(N)∣∣N1

' 1

4N√

3exp

(2π

√N

6


B ) · p(E−B )


S = ln p(E+B ) + ln p(E−B ) = 2π

(√c∆+

6+

√c∆−

6

)+ . . .



Microstate counting






p(N)∣∣N1

' 1

4N√

3exp

(2π

√N

6


B ) · p(E−B )


S = ln p(E+B ) + ln p(E−B ) = 2π

(√c∆+

6+

√c∆−

6

)+ . . .



Outline

Motivation



Discussion

Daniel Grumiller — Soft Heisenberg Hair Discussion 24/32

Relations to previous approaches

I Carlip ’94: conceptually very close; main difference: Carlip stressesnear horizon Virasoro, while we exploit near horizon Heisenberg

I Strominger, Vafa ’96: string construction of microstates that relies onBPS/extremality

I Strominger ’97: exploits Cardy formula for microstate counting, butdoes not identify microstates

I Mathur ’05: fuzzballs, like horizon fluff, do not have horizon; we needonly semi-classical near horizon description, not full quantum gravity

I Donnay, Giribet, Gonzalez, Pino ’15: near horizon bc’s (fixed Rindler)with centerless warped algebra

I Afshar, Detournay, DG, Oblak ’15: near horizon bc’s (varying Rindler)and null compactification with twisted warped algebra

I Hawking, Perry, Strominger ’16: introduced notion of “soft hair”, butdid not attempt microstate counting

I Afshar, Detournay, DG, Merbis, Perez, Tempo, Troncoso ’16:introduced near horizon bc’s we use; did not attempt construction ofmicrostates (but does Cardy-type of counting)








































































Generalization to four dimensions

Compare with near horizon construction of Donnay, Giribet, Gonzalez, Pino ’15

I Near horizon algebra similar to but different from BT-BMS4:

[Y±n , Y±m] = (n−m)Y±n+m

[Y+l , T(n,m)] = −n T(n+l,m)

[Y−l , T(n,m)] = −m T(n,m+l)

I Intriguing algebraic observation: introducing again

[J ±n , J ±m ] = 12 n δn,−m = −[K±n , K±m]

recovers 4d algebra above by “Sugawara construction”

T(n,m) =(J +n +K+

n

)(J −m +K−m

)Y±n =

∑p∈Z

(J ±n−p +K±n−p

)(J ±p −K±p

)I Making AKVs in DGGP state-dependent to leading order relates their

canonical boundary charges to Heisenberg boundary chargesI Indicates existence of soft Heisenberg hair in 4d


Generalization to four dimensionsCompare with near horizon construction of Donnay, Giribet, Gonzalez, Pino ’15


[Y±n , Y±m] = (n−m)Y±n+m

[Y+l , T(n,m)] = −n T(n+l,m)

[Y−l , T(n,m)] = −m T(n,m+l)


[J ±n , J ±m ] = 12 n δn,−m = −[K±n , K±m]


T(n,m) =(J +n +K+

n

)(J −m +K−m

)Y±n =

∑p∈Z


)(J ±p −K±p






[Y±n , Y±m] = (n−m)Y±n+m

[Y+l , T(n,m)] = −n T(n+l,m)

[Y−l , T(n,m)] = −m T(n,m+l)


[J ±n , J ±m ] = 12 n δn,−m = −[K±n , K±m]


T(n,m) =(J +n +K+

n

)(J −m +K−m

)Y±n =

∑p∈Z


)(J ±p −K±p

)

I Making AKVs in DGGP state-dependent to leading order relates theircanonical boundary charges to Heisenberg boundary charges

I Indicates existence of soft Heisenberg hair in 4d




[Y±n , Y±m] = (n−m)Y±n+m

[Y+l , T(n,m)] = −n T(n+l,m)

[Y−l , T(n,m)] = −m T(n,m+l)


[J ±n , J ±m ] = 12 n δn,−m = −[K±n , K±m]


T(n,m) =(J +n +K+

n

)(J −m +K−m

)Y±n =

∑p∈Z


)(J ±p −K±p


canonical boundary charges to Heisenberg boundary charges

I Indicates existence of soft Heisenberg hair in 4d




[Y±n , Y±m] = (n−m)Y±n+m

[Y+l , T(n,m)] = −n T(n+l,m)

[Y−l , T(n,m)] = −m T(n,m+l)


[J ±n , J ±m ] = 12 n δn,−m = −[K±n , K±m]


T(n,m) =(J +n +K+

n

)(J −m +K−m

)Y±n =

∑p∈Z


)(J ±p −K±p




Microstates of non-extremal Kerr?

Main challenge: how to provide (controlled) cut-off on soft hair spectrumin four dimensions?


Thanks for your attention!

H. Afshar, D. Grumiller and M.M. Sheikh-Jabbari “Near Horizon SoftHairs as Microstates of Three Dimensional Black Holes,” 1607.00009.

H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez,D. Tempo and R. Troncoso “Soft Heisenberg hair on black holes inthree dimensions,” Phys.Rev. D93 (2016) 101503(R); 1603.04824.

Thanks to Bob McNees for providing the LATEX beamerclass!




Map to asymptotic variables

I Usual asymptotic AdS3 connection with chemical potential µ:

A = b−1(

d+a)b aϕ = L+ − 1

2 LL−b = eρL0 at = µL+ − µ′L0 +

(12 µ′′ − 1

2 Lµ)L−

I Gauge trafo a = g−1 (d+a) g with

g = exp (xL+) · exp (−12JL−)

where ∂vx− ζx = µ and x′ − J x = 1I Near horizon chemical potential transforms into combination of

asymptotic charge and chemical potential!

µ′ − J µ = −ζI Asymptotic charges: twisted Sugawara construction with near horizon

chargesL = 1

2J2 + J ′

I Get Virasoro with non-zero central charge δL = 2Lε′ + L′ε− ε′′′




A = b−1(

d+a)b aϕ = L+ − 1

2 LL−b = eρL0 at = µL+ − µ′L0 +

(12 µ′′ − 1

2 Lµ)L−


g = exp (xL+) · exp (−12JL−)

where ∂vx− ζx = µ and x′ − J x = 1

I Near horizon chemical potential transforms into combination ofasymptotic charge and chemical potential!


chargesL = 1

2J2 + J ′





A = b−1(

d+a)b aϕ = L+ − 1

2 LL−b = eρL0 at = µL+ − µ′L0 +

(12 µ′′ − 1

2 Lµ)L−


g = exp (xL+) · exp (−12JL−)



µ′ − J µ = −ζ

I Asymptotic charges: twisted Sugawara construction with near horizoncharges

L = 12J

2 + J ′





A = b−1(

d+a)b aϕ = L+ − 1

2 LL−b = eρL0 at = µL+ − µ′L0 +

(12 µ′′ − 1

2 Lµ)L−


g = exp (xL+) · exp (−12JL−)




chargesL = 1

2J2 + J ′





A = b−1(

d+a)b aϕ = L+ − 1

2 LL−b = eρL0 at = µL+ − µ′L0 +

(12 µ′′ − 1

2 Lµ)L−


g = exp (xL+) · exp (−12JL−)




chargesL = 1

2J2 + J ′



Remarks on asymptotic and near horizon variables

I Asymptotic spin-2 currents fulfill Virasoro algebra, but charges obeystill Heisenberg algebra

δQ = − k

4π

∮dϕε δL = − k

4π

∮dϕη δJ

Reason: asymptotic “chemical potentials” µ depend on near horizoncharges J and chemical potentials ζ

I Our boundary conditions singled out: whole spectrum compatiblewith regularity

I For constant chemical potential ζ: regularity = holonomy condition

µµ′′ − 12µ′ 2 − µ2L = −2π2/β2

Solved automatically from map to asymptotic observables; reminder:

µ′ − J µ = −ζ L = 12J

2 + J ′

Near horizon boundary conditions natural for near horizon observer




δQ = − k

4π


4π

∮dϕη δJ




µµ′′ − 12µ′ 2 − µ2L = −2π2/β2


µ′ − J µ = −ζ L = 12J

2 + J ′





δQ = − k

4π


4π

∮dϕη δJ




µµ′′ − 12µ′ 2 − µ2L = −2π2/β2


µ′ − J µ = −ζ L = 12J

2 + J ′





δQ = − k

4π


4π

∮dϕη δJ




µµ′′ − 12µ′ 2 − µ2L = −2π2/β2


µ′ − J µ = −ζ L = 12J

2 + J ′



On compatibility with AdS3/CFT2

Punchline: our proposal is Bohr-type quantization of spectrum

I Unitary representations of Virasoro algebra span Hilbert space HVir

I Gravity side, asymptotically: HVir splits into HBTZ, HConic and HAdS

I Only states in HBTZ captured by twisted Sugawara constructionI States in HBTZ labelled by two positive numbers (L±0 ) and set of

integers (Virasoro excitations)I Gravity side, near horizon: soft hair Hilbert space HNH includes all

soft hair descendants of vacuum labeled by set of integersI HNH contained in HVir

I CFT side: expected to have discrete set of primariesI HCFT contained in HVir

I Our proposal: HNH = HCFT

I Example ([Lunin], Maldacena, Maoz ’02): states in HCFT

corresponding to conic spaces discrete family L±0 = −rk/4 (r = 0:massless BTZ, r = k: global AdS, 0 < r < k: spectral flow)

I Spectral flow and discrete conic spaces generated by J ±r(r = 1, 2, . . . c− 1), the “horizon fluff”



















I Only states in HBTZ captured by twisted Sugawara construction

I States in HBTZ labelled by two positive numbers (L±0 ) and set ofintegers (Virasoro excitations)

I Gravity side, near horizon: soft hair Hilbert space HNH includes allsoft hair descendants of vacuum labeled by set of integers

I HNH contained in HVir












integers (Virasoro excitations)

I Gravity side, near horizon: soft hair Hilbert space HNH includes allsoft hair descendants of vacuum labeled by set of integers














soft hair descendants of vacuum labeled by set of integers




























I CFT side: expected to have discrete set of primaries

I HCFT contained in HVir


























































On log corrections

I Semi-classicallyS = S0 + # · lnS0 +O(1)

I Carlip ’00:

SBTZ = SBH −3

2· lnSBH +O(1)

I Naive application of Hardy–Ramanujan

S = SBH − 2 · lnSBH +O(1)

I Mismatch in coefficients; not sure yet if bug or feature


On log corrections


I Carlip ’00:

SBTZ = SBH −3

2· lnSBH +O(1)





On log corrections


I Carlip ’00:

SBTZ = SBH −3

2· lnSBH +O(1)





On log corrections


I Carlip ’00:

SBTZ = SBH −3

2· lnSBH +O(1)





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Soft Heisenberg Hair - quark.itp.tuwien.ac.atquark.itp.tuwien.ac.at/~grumil/pdf/nordita2017.pdf · Soft Heisenberg Hair Daniel Grumiller Institute for Theoretical Physics TU Wien