Probes in AdS 3 Quantum Gravity
403
5.1 Wilson Lines in AdS 3 /CFT 2
The Wilson loop in our 3D higher spin gravity is given by
W R (C) = Tr R
P exp
C
A
P exp
C
¯
A
.
(39)
Here A and ¯
A are the connections representing a higher spin background in
SL(N, R) Chern–Simons theory. The representation R is the infinite-dimensional
highest-weight representation of sl(N, R), and C is a loop in the bulk. We may also
consider an open-ended Wilson line operator. To define this object we specify the
locations of its endpoints (x i , x f ). We must also specify boundary data in the form
of two specific states |U i |U f ∈ R at these endpoints. The Wilson line operator
is then
W R (x i , x f ) = =U f |P exp
−
γ
A
P exp
−
γ
¯
A
|U i ,
(40)
where now γ (s) is a curve with bulk endpoints (x i , x f ) parametrized by s.
W R (x i , x f ) is no longer fully gauge-invariant; clearly it depends in a gaugecovariant manner on the choice of boundary data |U i |U f . Nevertheless, for flat
connections, W R (x i , x f ) only depends on the topology of γ , but not on the shape
of the curve.
In AdS 3 the presence of a boundary allows the formulation of suitably diffeomorphism invariant observables—the correlation functions of the dual CFT 2 —and
thus one would expect that it would be possible to compute such objects in
the Chern–Simons formulation. Some progress in this direction was made in
[16, 34, 38, 39, 47], motivated largely by the computation of entanglement entropy
of field theories dual to 3d bulk higher spin gravity. These developments have shown
that a Wilson line in an infinite-dimensional highest-weight representation R under
the bulk SL(N) × SL(N) gauge group could be used to compute boundary theory
correlators, i.e.
W R (x i , x f ) =
r→∞
|O(y i )O(y f )|Ψ ,
(41)
where we have picked coordinates x μ = (r, y i ) with r an AdS holographic coordinate and y i a CFT coordinate. See Fig. 2. Here the Wilson line W R ends on the
boundary at r → ∞, and Ψ denotes the CFT 2 state dual to a particular configuration
of Chern–Simons gauge fields that constitute the gravitational background in the
interior. The representation space R was generated from the Hilbert space of an
auxiliary SL(N)-valued quantum mechanical degree of freedom U(s) that lives on
the Wilson line. The quadratic Casimirs of the representation R mapped in the usual
manner to the conformal dimensions (h, ¯
h) of the dual CFT operator O(y).
403
5.1 Wilson Lines in AdS 3 /CFT 2
The Wilson loop in our 3D higher spin gravity is given by
W R (C) = Tr R
P exp
C
A
P exp
C
¯
A
.
(39)
Here A and ¯
A are the connections representing a higher spin background in
SL(N, R) Chern–Simons theory. The representation R is the infinite-dimensional
highest-weight representation of sl(N, R), and C is a loop in the bulk. We may also
consider an open-ended Wilson line operator. To define this object we specify the
locations of its endpoints (x i , x f ). We must also specify boundary data in the form
of two specific states |U i |U f ∈ R at these endpoints. The Wilson line operator
is then
W R (x i , x f ) = =U f |P exp
−
γ
A
P exp
−
γ
¯
A
|U i ,
(40)
where now γ (s) is a curve with bulk endpoints (x i , x f ) parametrized by s.
W R (x i , x f ) is no longer fully gauge-invariant; clearly it depends in a gaugecovariant manner on the choice of boundary data |U i |U f . Nevertheless, for flat
connections, W R (x i , x f ) only depends on the topology of γ , but not on the shape
of the curve.
In AdS 3 the presence of a boundary allows the formulation of suitably diffeomorphism invariant observables—the correlation functions of the dual CFT 2 —and
thus one would expect that it would be possible to compute such objects in
the Chern–Simons formulation. Some progress in this direction was made in
[16, 34, 38, 39, 47], motivated largely by the computation of entanglement entropy
of field theories dual to 3d bulk higher spin gravity. These developments have shown
that a Wilson line in an infinite-dimensional highest-weight representation R under
the bulk SL(N) × SL(N) gauge group could be used to compute boundary theory
correlators, i.e.
W R (x i , x f ) =
r→∞
|O(y i )O(y f )|Ψ ,
(41)
where we have picked coordinates x μ = (r, y i ) with r an AdS holographic coordinate and y i a CFT coordinate. See Fig. 2. Here the Wilson line W R ends on the
boundary at r → ∞, and Ψ denotes the CFT 2 state dual to a particular configuration
of Chern–Simons gauge fields that constitute the gravitational background in the
interior. The representation space R was generated from the Hilbert space of an
auxiliary SL(N)-valued quantum mechanical degree of freedom U(s) that lives on
the Wilson line. The quadratic Casimirs of the representation R mapped in the usual
manner to the conformal dimensions (h, ¯
h) of the dual CFT operator O(y).