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Higher-spin charges in Hamiltonian form. I. Bose fields
Received: August
Higher-spin charges in Hamiltonian form. I. Bose
0 Open Access , c The Authors
1 Casilla 1469 , Valdivia , Chile
2 ULB-Campus Plaine CP231 , 1050 Brussels , Belgium
We study asymptotic charges for symmetric massless higher-spin Anti de Sitter backgrounds of arbitrary dimension within the canonical formalism. We rst analyse in detail the spin-3 example: we cast Fronsdal's action in Hamiltonian form, we derive the charges and we propose boundary conditions on the canonical variables that secure their niteness. We then extend the computation of charges and the characterisation of boundary conditions to arbitrary spin.
Gauge Symmetry; Higher Spin Gravity; Field Theories in Higher Dimensions
Contents
1 Introduction 2 Spin-3 example 3
Arbitrary spin
Hamiltonian and constraints
Gauge transformations
Boundary conditions
Asymptotic symmetries
Constraints and gauge transformations
Boundary conditions and asymptotic symmetries
A Notation and conventions
Hamiltonian form of Fronsdal's action
B.1 Spin 3
B.2 Arbitrary spin
C Covariant boundary conditions
C.1 Fallo
of the solutions of the free equations of motion
C.2 Residual gauge symmetry
C.3 Initial data at the boundary
D Conformal Killing tensors
E Spin-s charges
Introduction
original literature).
Consequently, they bring new surface charges in the analysis of the dynamics.
constant-curvature [19{21] and at backgrounds [22, 23].
acting as an infrared regulator (see e.g. [11] for a review).
generalisations thereof.
correspondences.
by a spin-3
We detail the Hamiltonian description of the free dynamics and we
provide boundary conditions on the canonical variables that secure
niteness of charges.
We conclude summarising our results and discussing their expected regime of
more details on various results used in the main text (appendices B, D and E).
Spin-3 example
form the Fronsdal action for a spin-3
eld on an Anti de Sitter background of dimension
of gauge transformations and we propose boundary conditions on
elds and deformation
parameters that secure their
niteness. The
nal expression for the charges is given in
section 2.5, where we also compare our outcome with other results in literature.
We begin with the manifestly covariant Fronsdal action [42, 43]
S =
signal a trace, e.g. '
= '
. If one parameterises the AdSd background with static
ds2 =
f 2(xk)dt2 + gij (xk)dxidxj ;
one nds that the terms in the Lagrangian with two time derivatives are
L = pg
'_ ijk ('_ ijk
3 gij '_ k) +
the combination ('000
3f 2'0), while the remaining components of the symmetric tensor
are Lagrange multipliers which enforce rst class constraints. That ('000
the discussion around (2.19)).
It is convenient to perform the rede nitions
Introducing then the conjugate momenta to 'ijk and ,
L =
3 g(ij '_ k)
3 r(iN jk) + 3 g(ijj 2 r N jk) + @jk)N
one can equivalently rewrite the action (2.1) in Hamiltonian form as3
S['ijk; ; ijk; ~ ; N i; N ij ] =
2All results of this subsection apply also to de Sitter provided that one maps L ! iL.
3The rewriting of Fronsdal's action in Hamiltonian form has been previously discussed in [29],
relydescription of the free higher-spin dynamics on AdS.
where H, Ci and Cij are functions only of 'ijk,
and of their conjugate momenta
Christo el symbol depends on g00 as
0i = f 1@if :
be of constant curvature (see appendix B.1 for details).
The Hamiltonian in (2.7) reads explicitly
H = f
and a generalisation of the constraint that generates spatial di eomorphisms,
Cij =
Ci = 3 @i ~
r r 'i) + rir '
2 ri'jklrj 'ikl + 3 r 'ij ri'j
note here the dependence on the additional momentum ~ ),
the correct counting of local degrees of freedom (see e.g. x 1.4.2 of [47]):
# d.o.f. =
(d + 1)!
1) =
2)(d + 2)
in d space-time dimensions.
These correspond to the variation
= 3 r(
that, in the covariant language, leaves the action (2.1) invariant provided that
be traceless.
In the canonical formalism, the generator of gauge transformations is
[ ij ; i] =
well de ned functional derivatives, i.e. that its variation be again a bulk integral:
G =
ijk + Bijk 'ijk + C ~ + D
From (2.15) one can read the gauge transformations of the canonical variables as
'ijk = f'ijk; G[ ; ]g = Aijk ;
= f ; G[ ; ]g = C ;
ijk = f
ijk; G[ ; ]g =
~ = f ~ ; G[ ; ]g =
serving the given boundary conditions.
account that the background has constant curvature, one obtains
; (2.17b)
Aijk = 3 r(i jk) ;
Bijk = 3pg
C =
D =
in the form (2.15) must be cancelled by the variations
Q1 and Q2 of the charges. Being
linear in the elds, these variations are integrable and yield:4
Q1[ ij ] = 3 d
Q2[ i] = 3 d
dd 2x n^i, where n^i and dd 2x are respectively the normal and the product of di erentials
of the coordinates on the d
the determinant of the intrinsic metric that appears in the full volume element).
vanish for the zero solution.
parameter, while i is related to
i =
thanks to the Fronsdal constraint g
tives in the gauge variation of ('000
gauge transformations of the Lagrange multipliers: (...truncated)