  We study the self-energy of a gravitating point particle in AdS$_3$, and
compare to operator dimensions in CFT$_2$. In particular, we compute the one
and two loop diagram contributions to the expectation value of an open Wilson
line in the SL(2,R)$\times$ SL(2,R) Chern-Simons formulation of AdS$_3$
gravity. This gives the two-point function of CFT primary operators to second
order in a large $c$ expansion, and hence yields the scaling dimension $h(j,c)$
as a function of the SL(2,R) spin $j$. Comparison to CFT is made in the context
of constructing Virasoro representations starting from representations of
SL(2,R) current algebra. Our Wilson line computations follow the framework
advanced recently by Fitzpatrick et. al., which is based on earlier work by H.
Verlinde. We encounter some renormalization scheme ambiguities at the two-loop
level which we are not able to fully resolve, hampering a definitive comparison
with CFT expressions at this order.
