Symmetry-Resolved Entanglement in AdS${}_3$/CFT${}_2$ coupled to $U(1)$ Chern-Simons Theory

TL;DR

This paper explores symmetry-resolved entanglement entropy in AdS${}_3$/CFT${}_2$ with $U(1)$ Chern-Simons coupling, establishing a holographic dual involving charged Wilson lines and deriving explicit formulas for various geometries.

Contribution

It introduces a new holographic method relating charged moments to Wilson lines and calculates symmetry-resolved entanglement entropy in multiple AdS${}_3$ geometries, confirming results via dual CFT analysis.

Findings

Symmetry-resolved entanglement entropy depends on geodesic length and Chern-Simons level.

The method applies to Poincaré, global AdS${}_3$, and conical defect geometries.

Results show equipartition of entanglement across charge sectors.

Abstract

We consider symmetry-resolved entanglement entropy in AdS${}_3$/CFT${}_2$ coupled to $U(1)$ Chern-Simons theory. We identify the holographic dual of the charged moments in the two-dimensional conformal field theory as a charged Wilson line in the bulk of AdS${}_3$, namely the Ryu-Takayanagi geodesic minimally coupled to the $U(1)$ Chern-Simons gauge field. We identify the holonomy around the Wilson line as the Aharonov-Bohm phases which, in the two-dimensional field theory, are generated by charged $U(1)$ vertex operators inserted at the endpoints of the entangling interval. Furthermore, we devise a new method to calculate the symmetry resolved entanglement entropy by relating the generating function for the charged moments to the amount of charge in the entangling subregion. We calculate the subregion charge from the $U(1)$ Chern-Simons gauge field sourced by the bulk Wilson line. We use our method to derive the symmetry-resolved entanglement entropy for Poincar\'e patch and global AdS${}_3$, as well as for the conical defect geometries. In all three cases, the symmetry resolved entanglement entropy is determined by the length of the Ryu-Takayanagi geodesic and the Chern-Simons level $k$, and fulfills equipartition of entanglement. The asymptotic symmetry algebra of the bulk theory is of $\hat{\mathfrak{u}}{(1)_k}$ Kac-Moody type. Employing the $\hat{\mathfrak{u}}{(1)_k}$ Kac-Moody symmetry, we confirm our holographic results by a calculation in the dual conformal field theory.