Defect loops in gauged Wess-Zumino-Witten models

TL;DR

This work analyzes loop defects in gauged Wess-Zumino-Witten models and the action of renormalization-group flows on them. By enforcing global and affine symmetries, the authors reduce the defect parameter space and identify holomorphic, $\widehat{H}$-invariant defects that descend to gauge-invariant coset defects in $G/H$. They develop a perturbative $1/k$ expansion to derive RG flow equations for defect couplings, locate fixed points, and show exact quantization of these fixed points, which then yield a universal mechanism for boundary flows. The key advance is deriving the generalized Affleck-Ludwig rule from universal defect flows and connecting holomorphic defects in the WZW model to coset model boundary dynamics, including a BRST/GKO correspondence. The results illuminate how defect monodromies and coset modular data govern boundary RG behavior in rational CFTs and provide a precise, scheme-dependent picture of the invariant subspaces governing these flows.

Abstract

We consider loop observables in gauged Wess-Zumino-Witten models, and study the action of renormalization group flows on them. In the WZW model based on a compact Lie group G, we analyze at the classical level how the space of renormalizable defects is reduced upon the imposition of global and affine symmetries. We identify families of loop observables which are invariant with respect to an affine symmetry corresponding to a subgroup H of G, and show that they descend to gauge-invariant defects in the gauged model based on G/H. We study the flows acting on these families perturbatively, and quantize the fixed points of the flows exactly. From their action on boundary states, we present a derivation of the "generalized Affleck-Ludwig rule, which describes a large class of boundary renormalization group flows in rational conformal field theories.

Figures (1)

• Figure 1: The pattern of flows for the two examples discussed in the text: $G=SU(2)$, $H=O(2)$ ( left) and $G=SU(2)\times SU(2)$, $H=SU(2)_{\rm diag}$ ( right). $\lambda$ and $\tilde{\lambda}$ are plotted respectively along the vertical and horizontal axis. The Kondo flows of the $G$-symmetric defects ($\lambda = \tilde{\lambda}$), analyzed in Bachas:2004sy, are shown in green. The flows along the $\widehat{H}$-invariant subspaces ($\lambda = 1$) are drawn in red. They descend to the Fredenhagen-Schomerus flows in the $G/H$ coset models. The symmetry $\tilde{\lambda} \rightarrow -\tilde{\lambda}$ corresponds to automorphisms of the algebras $\mathfrak{g}$, as explained in the text. Both cases have fixed points at $(0,0), (0, 1)$ and $(1, \pm 1)$. In the first example, the first two fixed points lie on the $\lambda$ axis, which is a line of marginal deformations.