Quantization of Wilson loops in Wess-Zumino-Witten models

TL;DR

The paper introduces a non-perturbative quantization of classical Wilson loops in WZW models by constructing central operators $z_\mu$ acting on the closed-string Hilbert space, with eigenvalues tied to the modular $S$-matrix via $\tfrac{S_{\mu\lambda}}{S_{0\lambda}}$. It then establishes a precise open/closed duality: certain boundary perturbations in the open-string picture are equivalent to inserting these central Wilson-loop operators in the closed-string picture, enabling exact analysis of boundary RG flows and fixed points. The authors derive the full spectrum of the quantized Wilson loops, show the resulting operators form a representation/fusion ring, and demonstrate their utility in identifying symmetry-breaking branes and their ground states, including a detailed application to su(2) subalgebras of minimal index. This framework provides a powerful, exact tool to map boundary perturbations to bulk-defect operators, yielding tractable calculations of boundary states and partition functions across dual pictures. The results illuminate brane dynamics in WZW backgrounds and offer a concrete mechanism to generate and connect conformal boundary conditions via defect operators.

Abstract

We describe a non-perturbative quantization of classical Wilson loops in the WZW model. The quantized Wilson loop is an operator acting on the Hilbert space of closed strings and commuting either with the full Kac-Moody chiral algebra or with one of its subalgebras. We prove that under open/closed string duality, it is dual to a boundary perturbation of the open string theory. As an application, we show that such operators are useful tools for identifying fixed points of the boundary renormalization group flow.