Higher Spin Rindler Holography
Daniel Grumiller
Institute for Theoretical Physics TU Wien
MIAPP programme ‘Higher Spin Theory and Duality’
M¨unchen, May 2016
based on work w. (H. Afshar, S. Detournay, W. Merbis),
Simple punchline Heisenberg algebra
[Xn, Pm] =i δn, m fundamental not only in quantum mechanics
but also in near horizon physics of (higher spin) gravity theories
Outline
Motivation
Near horizon boundary conditions for spin-2
Generalization to spin-N
Soft Heisenberg hair
Soft hairy black hole entropy
Concluding comments
Outline
Motivation
Near horizon boundary conditions for spin-2
Generalization to spin-N
Soft Heisenberg hair
Soft hairy black hole entropy
Concluding comments
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
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
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, ...)
warped CFT: Detournay, Hartman, Hofman ’12
Galilean CFT: Bagchi, Detournay, Fareghbal, Simon ’13; Barnich ’13
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
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
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
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 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
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
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
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 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
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 chemical potential: 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
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 Ifchemical potential: all states in theory have same (Unruh-)temperature
TU = a 2π suggestion in1511.08687
We make this choice in this talk!
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
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 Ifchemical potential: 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
Outline
Motivation
Near horizon boundary conditions for spin-2
Generalization to spin-N
Soft Heisenberg hair
Soft hairy black hole entropy
Concluding comments
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 (1ζL+)·exp (ρ2L−). (assume constantζ for simplicity)
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
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 (1ζL+)·exp (ρ2L−). (assume constantζ for simplicity)
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
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 (1ζL+)·exp (ρ2L−). (assume constantζ for simplicity)
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
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 (1ζL+)·exp (ρ2L−). (assume constantζ for simplicity)
Near horizon metric Using
gµν = 12
A+µ −A−µ
A+ν −A−ν
yields (f := 1 +ρ/(2a))
ds2=−2aρfdv2+ 2 dvdρ−2ωa−1 dϕdρ + 4ωρfdvdϕ+
γ2+2ρaf(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”
Near horizon metric Using
gµν = 12
A+µ −A−µ
A+ν −A−ν yields (f := 1 +ρ/(2a))
ds2=−2aρfdv2+ 2 dvdρ−2ωa−1 dϕdρ + 4ωρfdvdϕ+
γ2+2ρaf(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:
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”
Near horizon metric Using
gµν = 12
A+µ −A−µ
A+ν −A−ν yields (f := 1 +ρ/(2a))
ds2=−2aρfdv2+ 2 dvdρ−2ωa−1 dϕdρ + 4ωρfdvdϕ+
γ2+2ρaf(γ2−ω2) dϕ2 state-dependent functions J± =γ±ω, chemical potentialsζ±=−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”
Near horizon metric Using
gµν = 12
A+µ −A−µ
A+ν −A−ν yields (f := 1 +ρ/(2a))
ds2=−2aρfdv2+ 2 dvdρ−2ωa−1 dϕdρ + 4ωρfdvdϕ+
γ2+2ρaf(γ2−ω2) dϕ2 state-dependent functions J± =γ±ω, chemical potentialsζ±=−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”
Near horizon metric Using
gµν = 12
A+µ −A−µ
A+ν −A−ν yields (f := 1 +ρ/(2a))
ds2=−2aρfdv2+ 2 dvdρ−2ωa−1 dϕdρ + 4ωρfdvdϕ+
γ2+2ρaf(γ2−ω2) dϕ2 state-dependent functions J± =γ±ω, chemical potentialsζ±=−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”
Near horizon metric Using
gµν = 12
A+µ −A−µ
A+ν −A−ν yields (f := 1 +ρ/(2a))
ds2=−2aρfdv2+ 2 dvdρ−2ωa−1 dϕdρ + 4ωρfdvdϕ+
γ2+2ρaf(γ2−ω2) dϕ2 state-dependent functions J± =γ±ω, chemical potentialsζ±=−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”
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!
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!
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!
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 Jn±= k
4π I
dϕ einϕJ±(ϕ)
I Near horizon symmetry algebra Jn±, Jm±
=±12knδn+m,0
Jn+, Jm−
= 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=Jn+−Jn− yields Heisenberg algebra(with CasimirsX0,P0)
[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0 [Xn, Pm]=iδn,m ifn6= 0
Near horizon symmetry algebra
I Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos
I Expand charges in Fourier modes Jn±= k
4π I
dϕ einϕJ±(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)
I Near horizon symmetry algebra Jn±, Jm±
=±12knδn+m,0
Jn+, Jm−
= 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=Jn+−Jn− yields Heisenberg algebra(with CasimirsX0,P0)
[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0 [Xn, Pm]=iδn,m ifn6= 0
Near horizon symmetry algebra
I Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos
I Expand charges in Fourier modes Jn±= k
4π I
dϕ einϕJ±(ϕ)
I Near horizon symmetry algebra Jn±, Jm±
=±12knδn+m,0
Jn+, Jm−
= 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=Jn+−Jn− yields Heisenberg algebra(with CasimirsX0,P0)
[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0 [Xn, Pm]=iδn,m ifn6= 0
Near horizon symmetry algebra
I Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos
I Expand charges in Fourier modes Jn±= k
4π I
dϕ einϕJ±(ϕ)
I Near horizon symmetry algebra Jn±, Jm±
=±12knδn+m,0
Jn+, Jm−
= 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=Jn+−Jn− yields Heisenberg algebra(with CasimirsX0,P0)
[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0 [Xn, Pm]=iδn,m ifn6= 0
Near horizon symmetry algebra
I Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos
I Expand charges in Fourier modes Jn±= k
4π I
dϕ einϕJ±(ϕ)
I Near horizon symmetry algebra Jn±, Jm±
=±12knδn+m,0
Jn+, Jm−
= 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=Jn+−Jn− yields Heisenberg algebra(with CasimirsX0,P0)
[Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0 [Xn, Pm]=iδn,m ifn6= 0
Outline
Motivation
Near horizon boundary conditions for spin-2
Generalization to spin-N
Soft Heisenberg hair
Soft hairy black hole entropy
Concluding comments
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 Wn(2)=Ln andJ(s)=J
I Reminder: relevant part ofsl(N) algebra: [Ln, Wm(s)] = n(s−1)−m
Wn+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)
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 Wn(2)=Ln andJ(s)=J
I Reminder: relevant part ofsl(N) algebra:
[Ln, Wm(s)] = n(s−1)−m Wn+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)
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 Wn(2)=Ln andJ(s)=J
I Reminder: relevant part ofsl(N) algebra:
[Ln, Wm(s)] = n(s−1)−m Wn+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)
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 Wn(2)=Ln andJ(s)=J
I Reminder: relevant part ofsl(N) algebra:
[Ln, Wm(s)] = n(s−1)−m Wn+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
Xη(s)W0(s)
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 einϕJ(s)(ϕ)
I Their algebra is again the Heisenberg algebra
[(Xs)n,(Xt)m] = [(Ps)n,(Pt)m] = [(Xs)0,(Pt)n] = [(Ps)0,(Xt)n] = 0 [(Xs)m,(Pt)n] =iδs,tδm,n for m6= 0.
with similar redefinitions as before (Ps)0 =J(s) 0+ +J(s) 0− (Ps)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
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 einϕJ(s)(ϕ)
I Their algebra is again the Heisenberg algebra
[(Xs)n,(Xt)m] = [(Ps)n,(Pt)m] = [(Xs)0,(Pt)n] = [(Ps)0,(Xt)n] = 0 [(Xs)m,(Pt)n] =iδs,tδm,n for m6= 0.
with similar redefinitions as before (Ps)0 =J(s) 0+ +J(s) 0− (Ps)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
Outline
Motivation
Near horizon boundary conditions for spin-2
Generalization to spin-N
Soft Heisenberg hair
Soft hairy black hole entropy
Concluding comments
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
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
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
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
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
Outline
Motivation
Near horizon boundary conditions for spin-2
Generalization to spin-N
Soft Heisenberg hair
Soft hairy black hole entropy
Concluding comments
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
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
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
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 = 2π(J0++J0−)
calculated directly in Chern–Simons formulation (in spin-2 case:
S =A/(4GN))
I No contribution from soft hair charges or higher spin charges
I
Before addressing microstates consider map to aymptotic variables
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 = 2π(J0++J0−)
I No contribution from soft hair charges or higher spin charges
I
Before addressing microstates consider map to aymptotic variables
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 = 2π(J0++J0−)
I No contribution from soft hair charges or higher spin charges
I Suggestive that microstate counting should work
Before addressing microstates consider map to aymptotic variables
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 =2π(J0++J0−)
I No contribution from soft hair charges or higher spin charges
I Suggestive that microstate counting should work
Before addressing microstates consider map to aymptotic variables
Map to asymptotic variables in spin-2 case
I Usual asymptotic AdS3 connection with chemical potential µ:
Aˆ= ˆb−1 d+ˆaˆb ˆaϕ=L+−12LL−
ˆb=eρL0 ˆat=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
Map to asymptotic variables in spin-2 case
I Usual asymptotic AdS3 connection with chemical potential µ:
Aˆ= ˆb−1 d+ˆaˆb ˆaϕ=L+−12LL−
ˆb=eρL0 ˆat=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
Map to asymptotic variables in spin-2 case
I Usual asymptotic AdS3 connection with chemical potential µ:
Aˆ= ˆb−1 d+ˆaˆb ˆaϕ=L+−12LL−
ˆb=eρL0 ˆat=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
Map to asymptotic variables in spin-2 case
I Usual asymptotic AdS3 connection with chemical potential µ:
Aˆ= ˆb−1 d+ˆaˆb ˆaϕ=L+−12LL−
ˆb=eρL0 ˆat=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
Map to asymptotic variables in spin-2 case
I Usual asymptotic AdS3 connection with chemical potential µ:
Aˆ= ˆb−1 d+ˆaˆb ˆaϕ=L+−12LL−
ˆb=eρL0 ˆat=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
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
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
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
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
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 kLn=X
p∈Z
Jn−pJp+iknJn
I Starting fromHeisenberg algebra obtain semi-classically Virasoro algebra
[Ln, Lm] = (n−m)Ln+m+
I Usual Cardy formula yields Bekenstein–Hawking result SCardy = 2π
q
L+0 + 2π q
L−0 = 2π(J0++J0−) = A
4GN =SBH
I NeedJ0vac=ik/2 to get correct vacuum valueLvac0 =−k/4; with a=iget modular transformed line-element ds2 =ρ2dt+ dρ2−dϕ2 Precise numerical factor intwist term crucial for correct results