Probes in AdS 3 Quantum Gravity
405
Fig. 3 Wilson line with
endpoints in different
boundaries of the eternal
black hole background. Here
W R (x i , x f ) allows us to test
properties that rely on the
presence of the left and right
boundary
Right
Left
Singularity
sided eternal black hole “grows” with time (as measured by the action of a bulk
Wilson line). We also highlighted some interesting features of purely one-sided
correlators, in particular the behavior of the extremal limit and provided evidence
for the emergence of an infrared AdS 2 .
Finally, another interesting development is the results we obtained in [46]. By
selecting |U to be singlet states of so(2, 2), we showed that
W R (x i , x f ) =
e −2hD(x i ,x f )
1 − e −2D(x i ,x f )
(42)
where D(x i , x f ) is the geodesic length of an effective metric given by
g μν =
1
2
Tr(A μ − U
−1 ¯
A μ U )(A ν − U
−1 ¯
A ν U) .
(43)
Equation (42) is the familiar bulk-to-bulk propagator of a minimally coupled scalar
field in a locally AdS 3 background [48, 49]. An expression such as (42) makes rather
evident that the Wilson line is a propagator, and hence its ties to geometry. It should
be viewed as an operator that satisfies
1
2
x f − 2h(h − 1)
W R (x f , x i ) =
1
8π
δ(x f , x i )
√ −g
,
(44)
where now x f is the Laplacian on the bulk AdS 3 spacetime.
An important issue that we have not addressed is quantum corrections due to
fluctuations of the background connections. This would capture 1/c corrections, i.e.
corrections controlled by the AdS radius in Planck units, or equivalently subleading
terms controlled by the level of the Chern–Simons theory. Work in this direction
has been done for SL(2) Chern–Simons theory, where Virasoro conformal blocks
are known to be tied to appropriate Wilson line in Chern–Simons [50–52]. Recent
developments for this holomorphic theory include [53–59]. It would be interesting
to evaluate 1/c corrections of our worldline quantum mechanics; in this case we
405
Fig. 3 Wilson line with
endpoints in different
boundaries of the eternal
black hole background. Here
W R (x i , x f ) allows us to test
properties that rely on the
presence of the left and right
boundary
Right
Left
Singularity
sided eternal black hole “grows” with time (as measured by the action of a bulk
Wilson line). We also highlighted some interesting features of purely one-sided
correlators, in particular the behavior of the extremal limit and provided evidence
for the emergence of an infrared AdS 2 .
Finally, another interesting development is the results we obtained in [46]. By
selecting |U to be singlet states of so(2, 2), we showed that
W R (x i , x f ) =
e −2hD(x i ,x f )
1 − e −2D(x i ,x f )
(42)
where D(x i , x f ) is the geodesic length of an effective metric given by
g μν =
1
2
Tr(A μ − U
−1 ¯
A μ U )(A ν − U
−1 ¯
A ν U) .
(43)
Equation (42) is the familiar bulk-to-bulk propagator of a minimally coupled scalar
field in a locally AdS 3 background [48, 49]. An expression such as (42) makes rather
evident that the Wilson line is a propagator, and hence its ties to geometry. It should
be viewed as an operator that satisfies
1
2
x f − 2h(h − 1)
W R (x f , x i ) =
1
8π
δ(x f , x i )
√ −g
,
(44)
where now x f is the Laplacian on the bulk AdS 3 spacetime.
An important issue that we have not addressed is quantum corrections due to
fluctuations of the background connections. This would capture 1/c corrections, i.e.
corrections controlled by the AdS radius in Planck units, or equivalently subleading
terms controlled by the level of the Chern–Simons theory. Work in this direction
has been done for SL(2) Chern–Simons theory, where Virasoro conformal blocks
are known to be tied to appropriate Wilson line in Chern–Simons [50–52]. Recent
developments for this holomorphic theory include [53–59]. It would be interesting
to evaluate 1/c corrections of our worldline quantum mechanics; in this case we