U(1) CS Theory vs SL(2) CS Formulation: Boundary Theory and Wilson Line

TL;DR

This paper explores the boundary theories derived from U(1) and SL(2) Chern-Simons formulations, analyzing Wilson lines, entanglement entropy, and quantum corrections, revealing discrepancies in quantum effects and the role of the Hayward term.

Contribution

It provides a detailed comparison between U(1) and SL(2) Chern-Simons theories, introduces a quantum generalization of entanglement entropy, and clarifies the impact of quantum corrections on boundary actions.

Findings

U(1) Chern-Simons boundary action includes a self-interaction term.

Quantum correction in pure AdS$_3$ gravity shifts central charge by 26.

Hayward term's quantum correction is shown to be incorrect.

Abstract

We first derive the boundary theory from the U(1) Chern-Simons theory. The boundary action on an $n$-sheet manifold appears from its back-reaction of the Wilson line. The reason is that the U(1) Chern-Simons theory can provide an exact effective action when introducing the Wilson line. The Wilson line in the pure AdS$_3$ Einstein gravity is equivalent to entanglement entropy in the boundary theory up to classical gravity. The U(1) Chern-Simons theory deviates by a self-interaction term from the gauge formulation on the boundary. We also compare the Hayward term in the SL(2) Chern-Simons formulation to the Wilson line approach. Introducing two wedges can reproduce the entanglement entropy for a single interval at the classical level. We propose quantum generalization by combining the bulk and Hayward terms. The quantum correction of the partition function vanishes. In the end, we calculate the entanglement entropy for a single interval. The pure AdS$_3$ Einstein gravity theory shows a shift of central charge by 26 at the one-loop level. The U(1) Chern-Simons theory does not show a shift from the quantum effect. The result is the same in the weak gravitational constant limit. The non-vanishing quantum correction shows that the Hayward term is incorrect.