Anomalous dimensions from quantum Wilson lines

TL;DR

This paper calculates quantum corrections to operator dimensions in AdS$_3$/CFT$_2$ using Wilson lines in Chern-Simons theory, revealing insights into anomalous dimensions and highlighting renormalization ambiguities at two loops.

Contribution

It introduces a detailed computation of anomalous dimensions via Wilson lines in AdS$_3$ gravity, extending previous frameworks and addressing higher-order quantum effects.

Findings

Computed two-loop corrections to Wilson line expectation values.

Identified renormalization ambiguities at two loops.

Provided a comparison framework between bulk calculations and CFT operator dimensions.

Abstract

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.