Probes in AdS 3 Quantum Gravity
391
2 AdS 3 Higher Spin Gravity
The simplest way to craft a higher spin theory exploits the Chern–Simons formulation of three-dimensional gravity: general relativity with a negative cosmological
constant can be reformulated as a SL(2, R) × SL(2, R) Chern–Simons theory
[1, 2, 12]. A high spin theory can be crafted by simply taking instead SL(N, R) ×
SL(N, R), which will produce an interacting higher spin theory for symmetric
tensors of spin s = 2, 3, . . . , N [13]. There are of course other ways to build higher
spin theories, but here we restrict the attention to these models. For a more complete
discussion on properties of these theories, see, for example, [9, 14, 15].
The action of the SL(N, R) × SL(N, R) Chern–Simons theory is given by
S = S CS [A] − S CS [ ¯
A] , S CS [A] =
k
4π
M
tr
A ∧ dA +
2
3
A ∧ A ∧ A
.
(1)
Here M is the 3-manifold that supports the sl(N, R) algebra valued connections A
and ¯
A, and the trace “tr” denotes the invariant quadratic form of the Lie algebra.
The equations of motion following from (1) are
dA + A ∧ A = 0 , d ¯
A + ¯
A ∧ ¯
A = 0 .
(2)
The conventions here follow those in [16].
The metric and higher spin fields are obtained from the Chern–Simons connection as symmetric, traceless tensors that transform in the spin s representation of
SL(2, R). For example, the metric and the spin three fields can be expressed as
follows
g μν ∼ tr
e μ e ν
, φ μνρ ∼ tr
e (μ e ν e ρ)
,
(3)
where, in line with the pure gravity case, one defines
e =
2
A − ¯
A
, ω =
1
2
A + ¯
A
(4)
and we introduced the AdS radius . The metric and higher spin fields can then be
expressed in terms of trace invariants of the vielbein [6, 7], with the total number
of inequivalent invariants being N − 1 for sl(N, R). This definition for metric-like
fields is appropriate for the principal embedding of sl(2, R) in sl(N, R).
The relation between the Chern–Simons level and the gravitational couplings is
k =
8G 3 N
, , N ≡ tr f (L 0 L 0 ) =
1
12
N(N
2
− 1) ,
(5)
in accordance with the pure gravity limit. The notation tr f denotes a trace in
the fundamental representation of sl(N, R). The central charge of the asymptotic
symmetry group is [5, 6]
391
2 AdS 3 Higher Spin Gravity
The simplest way to craft a higher spin theory exploits the Chern–Simons formulation of three-dimensional gravity: general relativity with a negative cosmological
constant can be reformulated as a SL(2, R) × SL(2, R) Chern–Simons theory
[1, 2, 12]. A high spin theory can be crafted by simply taking instead SL(N, R) ×
SL(N, R), which will produce an interacting higher spin theory for symmetric
tensors of spin s = 2, 3, . . . , N [13]. There are of course other ways to build higher
spin theories, but here we restrict the attention to these models. For a more complete
discussion on properties of these theories, see, for example, [9, 14, 15].
The action of the SL(N, R) × SL(N, R) Chern–Simons theory is given by
S = S CS [A] − S CS [ ¯
A] , S CS [A] =
k
4π
M
tr
A ∧ dA +
2
3
A ∧ A ∧ A
.
(1)
Here M is the 3-manifold that supports the sl(N, R) algebra valued connections A
and ¯
A, and the trace “tr” denotes the invariant quadratic form of the Lie algebra.
The equations of motion following from (1) are
dA + A ∧ A = 0 , d ¯
A + ¯
A ∧ ¯
A = 0 .
(2)
The conventions here follow those in [16].
The metric and higher spin fields are obtained from the Chern–Simons connection as symmetric, traceless tensors that transform in the spin s representation of
SL(2, R). For example, the metric and the spin three fields can be expressed as
follows
g μν ∼ tr
e μ e ν
, φ μνρ ∼ tr
e (μ e ν e ρ)
,
(3)
where, in line with the pure gravity case, one defines
e =
2
A − ¯
A
, ω =
1
2
A + ¯
A
(4)
and we introduced the AdS radius . The metric and higher spin fields can then be
expressed in terms of trace invariants of the vielbein [6, 7], with the total number
of inequivalent invariants being N − 1 for sl(N, R). This definition for metric-like
fields is appropriate for the principal embedding of sl(2, R) in sl(N, R).
The relation between the Chern–Simons level and the gravitational couplings is
k =
8G 3 N
, , N ≡ tr f (L 0 L 0 ) =
1
12
N(N
2
− 1) ,
(5)
in accordance with the pure gravity limit. The notation tr f denotes a trace in
the fundamental representation of sl(N, R). The central charge of the asymptotic
symmetry group is [5, 6]