Renormalization of gravitational Wilson lines

TL;DR

This paper investigates the renormalization of gravitational Wilson lines in 2D conformal field theory, providing a perturbative approach that matches exact anomalous dimensions and explores related models in current algebra.

Contribution

It introduces a systematic regularization and renormalization scheme for gravitational Wilson lines and verifies perturbative results against exact predictions up to order 1/c^3.

Findings

Perturbative calculations reproduce the exact anomalous dimension to order 1/c^3.

A combined dimensional regularization and analytic continuation method is effective.

Distinction between holomorphic and non-holomorphic Wilson lines is clarified.

Abstract

We continue the study of the Wilson line representation of conformal blocks in two-dimensional conformal field theory; these have an alternative interpretation as gravitational Wilson lines in the context of the AdS$_3$/CFT$_2$ correspondence. The gravitational Wilson line involves a path-ordered exponential of the stress tensor, and its expectation value can be computed perturbatively in an expansion in inverse powers of the central charge $c$. The short-distance singularities which occur in the associated stress tensor correlators require systematic regularization and renormalization prescriptions, whose consistency with conformal Ward identities presents a subtle problem. The regularization used here combines dimensional regularization and analytic continuation. Representation theoretic arguments, based on SL(2,R) current algebra, predict an exact result for the Wilson line anomalous dimension and, by building on previous work, we verify that the perturbative calculations using our regularization and renormalization prescriptions reproduce the exact result to order $1/c^3$ included. We also discuss a related, but somewhat simpler, Wilson line in Wess-Zumino-Witten models that yields current algebra conformal blocks, and we emphasize the distinction between Wilson lines constructed out of non-holomorphic and purely holomorphic currents.