Anomalous dimensions from quantum Wilson lines

TL;DR

This work investigates the gravitational self-energy of a point particle in AdS$_3$ and its reflection in dual CFT$_2$ via an open Wilson line in SL(2,R)×SL(2,R) Chern-Simons gravity. By computing the Wilson line two-point function to order $1/c^2$, the authors extract the scaling dimension $h(j,c)$ as a function of the SL(2,R) spin $j$ and compare to Virasoro expectations arising from constraining SL(2,R) current algebra. They find exact matches at leading and subleading orders ($h_0(j)=-j$, $h_1(j)=-6j(j+1)$) but encounter renormalization ambiguities at order $1/c^2$ that prevent a definitive determination of $h_2(j)$, highlighting the need for a principled renormalization scheme or canonical quantization to realize Virasoro symmetry in the bulk. The results illuminate the bulk realization of conformal blocks and the delicate interplay between UV divergences and Virasoro structure in holographic Wilson-line constructions.

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.

Figures (3)

• Figure 1: Wilson line diagrams to order $1/c^2$
• Figure 2: Graviton self energy
• Figure 3: Diagrams contributing at order $1/c^2$, with stress tensor insertions on the Wilson line as indicated.