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Renormalization of gravitational Wilson lines

Mert Besken, Eric D'Hoker, Ashwin Hegde, Per Kraus

TL;DR

The paper develops a controlled perturbative framework for the gravitational Wilson line in a 2D CFT, showing that dimensional regularization with carefully chosen renormalization constants reproduces the exact Virasoro vacuum block dimension $h(j,c)$ through order $1/c^3$. It presents both holomorphic and non-holomorphic Wilson lines in the WZW context, confirming that their anomalous dimensions match the current-algebra predictions and connecting to bi-local primary operators. A second regulator in strictly 2D fails to restore conformal invariance, underscoring subtle renormalization issues and suggesting possible operator-valued refinements and Hamiltonian-reduction interpretations. The work clarifies when and how Virasoro symmetry can control nonperturbative gravity corrections in AdS$_3$/CFT$_2$ via Wilson lines. Overall, it advances a coherent, regulator-sensitive program to extract conformal data from gravitational Wilson lines and exposes deep links between bulk gravity, current algebra, and Virasoro blocks.

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.

Renormalization of gravitational Wilson lines

TL;DR

The paper develops a controlled perturbative framework for the gravitational Wilson line in a 2D CFT, showing that dimensional regularization with carefully chosen renormalization constants reproduces the exact Virasoro vacuum block dimension through order . It presents both holomorphic and non-holomorphic Wilson lines in the WZW context, confirming that their anomalous dimensions match the current-algebra predictions and connecting to bi-local primary operators. A second regulator in strictly 2D fails to restore conformal invariance, underscoring subtle renormalization issues and suggesting possible operator-valued refinements and Hamiltonian-reduction interpretations. The work clarifies when and how Virasoro symmetry can control nonperturbative gravity corrections in AdS/CFT via Wilson lines. Overall, it advances a coherent, regulator-sensitive program to extract conformal data from gravitational Wilson lines and exposes deep links between bulk gravity, current algebra, and Virasoro blocks.

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/CFT 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 . 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 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.

Paper Structure

This paper contains 43 sections, 164 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organization
  3. The gravitational Wilson line operator
  4. Wilson line covariant under global conformal transformations
  5. Wilson line covariant under local conformal transformations
  6. Current algebra Wilson lines in the WZW model
  7. WZW model and current algebra
  8. Holomorphic Wilson line
  9. Lowest order calculations
  10. Higher order computations
  11. Non-holomorphic Wilson line from WZW
  12. Direct perturbative computation of $\langle g^{-1}(x_2) g(x_1)\rangle$
  13. Non-holomorphically factorized Wilson line
  14. Comments on holomorphic factorization
  15. Renormalization of gravitational Wilson lines
  16. ...and 28 more sections

Figures (1)

  • Figure 1: Feynman diagrams at order $1/k^2$.