Charges in Nonlinear Higher-Spin Theory
(V.E. Didenko, N.M., M.A. Vasiliev, JHEP 03 (2017) 164)
Nikita Misuna
Moscow Institute of Physics and Technology
Ginzburg Conference, Moscow, 23/05/17
Nikita Misuna (Moscow Institute of Physics and Technology)
Charges in Nonlinear Higher-Spin Theory
Ginzburg Conference, Moscow, 23/05/17
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Outline
• Nonlinear HS theory
• AdS4 charges
• Black Holes in HS Theory
• HS BH Charges
Nikita Misuna (Moscow Institute of Physics and Technology)
Charges in Nonlinear Higher-Spin Theory
Ginzburg Conference, Moscow, 23/05/17
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Nonlinear HS Equations
• Nonlinear HS theory is described by Vasiliev equations [Vasiliev’ 92] –
generating equations on HS fields, described by spacetime 1-forms ω (Y|x)
and 0-forms C (Y|x), polynomial in sp (4)-spinors Y A ;
• HS equations have schematic form
R := dω + ω ∗ ω = Υ (ω, ω, C) + Υ (ω, ω, C, C) + ...
dC + [ω, C]∗ = Υ (ω, C, C) + ...
where vertices Υ (...) have to be determined from Vasiliev equations.
• sp (4)-spinors Y A form HS algebra with following product:
f (Y) ∗ g (Y) =
d 4 Ud 4 V exp iUA V A f (Y + U) g (y + V)
A B
Y , Y ∗ = 2iAB ,
trf (Y) := f (0) ;
Nikita Misuna (Moscow Institute of Physics and Technology)
tr (f ∗ g) = tr (g ∗ f) .
Charges in Nonlinear Higher-Spin Theory
Ginzburg Conference, Moscow, 23/05/17
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AdS4 vacuum
• Vacuum:
C = 0,
dΩ + Ω ∗ Ω = 0.
• Symmetries:
δΩ = d + [Ω, ]∗ ;
D0 := d + [Ω0 , ]∗ = 0
– leftover global symmetries for fixed Ω0 .
• Natural choice is AdS4 -connection (so (3, 2) ≈ sp (4))
Ω0 = ΩAB
0 YA YB :=
i αβ
ωL yα yβ + ω̄Lα̇β̇ ȳα̇ ȳβ̇ + 2λeαβ̇ yα ȳβ̇ ,
4
D0 K AB YA YB = 0;
K AB (x) – usual Killing vector along with its covariant derivative.
Nikita Misuna (Moscow Institute of Physics and Technology)
Charges in Nonlinear Higher-Spin Theory
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Physical and topological HS fields
• Perturbative expansion over AdS4 background results in two different
representations for HS fields:
• adjoint representation
D0 f (Y|x) = df + [ωL + e, f]∗ ;
• twisted-adjoint representation
e 0 f (Y|x) = df + [ωL , f] + {e, f} ;
D
∗
∗
• This leads at the free level to the splitting into topological and physical
sectors:
• physical
e 0 C phys = 0;
D
• topological
D0 C top = 0;
• At the interaction level sectors become entangled.
Nikita Misuna (Moscow Institute of Physics and Technology)
Charges in Nonlinear Higher-Spin Theory
Ginzburg Conference, Moscow, 23/05/17
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AdS4 asymptotic charges
• GR asympt. charge for AdS-space [Ashtekar, Das ’99]:
Eab ξ a dSb ,
Q = const ·
C
a
ξ – asympt. symm. parameter, Eab – ’electric’ part of Weyl tensor.
• Analogous construction can be built in HS theory. From R := dω + ω ∗ ω it
follows
DR := dR + [ω, R]∗ = 0.
• If asymptotically (xz → 0)
ω (Y|x) → Ω0 ,
C (Y|x) → 0,
J := tr ( ∗ R) ,
D → D0 = 0,
dJ → 0.
Q =
J
Σ2
– asymptotically conserved HS charge.
Nikita Misuna (Moscow Institute of Physics and Technology)
Charges in Nonlinear Higher-Spin Theory
Ginzburg Conference, Moscow, 23/05/17
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AdS4 HS charges
• Nonlinear HS theory allows a completion of the asympt. charge to the bulk:
D0 = 0
holds at the free level for topological fields = C top .
• In higher orders C top get nonlinear corrections and source phys. fields:
DC top = Υ ω, C top , C top + Υ (ω, C, C) + ...
R := dω + ω ∗ ω = Υ ω, ω, C, C top + ...
• This permits to define
J := tr (R) ,
Q :=
J,
Σ2
Z := exp −
and to treat
J C top
Σ2
as a partition and C
top
Nikita Misuna (Moscow Institute of Physics and Technology)
as chemical potentials conjugated to HS charges.
Charges in Nonlinear Higher-Spin Theory
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HS AdS4 Black Holes
• BH metric in Kerr-Schild form [Carter ’68]
AdS
gmn = gmn
+
2M
km kn .
r
• Kerr-Schild vectors km :
km km = 0,
kn Dn0 km = 0,
1
1
= − Dm
km .
r
2 0
• HS generalisation [Didenko, Matveev, Vasiliev ’08]:
φm1 ...ms =
2M
km1 ...kms
r
obeys free spin-s equation in AdS.
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Charges in Nonlinear Higher-Spin Theory
Ginzburg Conference, Moscow, 23/05/17
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HS AdS4 Black Holes
• km = km (KAB ) – generic AdS4 BH is completely determined by global
symmetry KAB of empty AdS4 [Didenko, Matveev, Vasiliev ’08, ’09]
• HS BH Weyl tensors [Didenko, Vasiliev ’09]:
Cα(2s) =
q = 2−3/2 λ−2
ms λ−2s
s
(Kαα )
s!2s q2s+1
C̄α̇(2s) =
s
m̄s λ−2s
K̄
,
α̇
α̇
s!2s q̄2s+1
p
−Kαβ K αβ .
• Kerr-like HS BH is generated by
metric with rotation parameter a.
Nikita Misuna (Moscow Institute of Physics and Technology)
∂
= (1, 0, 0, 0) in AdS4 Boyer-Lindquist
∂t
Charges in Nonlinear Higher-Spin Theory
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Charge for Kerr-like HS BH
• Free equations generate linear vacuum partition:
D0 ω (y, ȳ|x) = −
iη α̇ β α̇ ∂ 2
i η̄ α αβ̇ ∂ 2
eβ e
e e
C
(0,
ȳ|x)
−
C (y, 0|x)
4
∂ȳα̇ ∂ȳα̇
4 β̇
∂yα ∂yα
• To the vacuum partition at the free level only spin-1 contributes:
J0 = −
iη α̇ β α̇
i η̄
eβ e Cα̇α̇ (x) − eα β̇ eαβ̇ Cαα (x)
4
4
Nikita Misuna (Moscow Institute of Physics and Technology)
Q0 = 4π
m1 η̄ − m̄1 η
.
1 + Λa2
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Charges in Nonlinear Higher-Spin Theory
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First order topological contribution
• First order corrections:
i
D0 ω (y, ȳ|x) = −
4
µeβ α̇ eβ α̇
∂2 top
C, C
(0, ȳ|x) + h.c.
∗
∂ȳα̇ ∂ȳα̇
• Spin-2 sector
C top = KAB (x) Y A Y B ,
D0 C top = 0,
contribution to spin-2 charge is
1
Js=2
=−
Nikita Misuna (Moscow Institute of Physics and Technology)
iµ α̇ β α̇ α̇α̇
eβ e K̄ Cα̇(4) (x) + h.c.
2
Q1s=2 = 3πΛ
m2 µ̄ − m̄2 µ
.
1 + Λa2
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Charges in Nonlinear Higher-Spin Theory
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Conclusion
• A generalisation of Ashtekar-Das asymptotic charge for HS theory is
proposed, allowing a nonlinear bulk completion;
• From thermodynamical perspective, topological fields play a role of chemical
potentials in HS partition;
• Free vacuum contribution to the partition is calculated, agreed with
ADM-like behaviour;
• Possible relations to the BH entropy and information paradox?
Nikita Misuna (Moscow Institute of Physics and Technology)
Ginzburg Conference, Moscow, 23/05/17
Charges in Nonlinear Higher-Spin Theory
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