Loop Operators and the Kondo Problem

Abstract

We analyse the renormalisation group flow for D-branes in WZW models from the point of view of the boundary states. To this end we consider loop operators that perturb the boundary states away from their ultraviolet fixed points, and show how to regularise and renormalise them consistently with the global symmetries of the problem. We pay particular attention to the chiral operators that only depend on left-moving currents, and which are attractors of the renormalisation group flow. We check (to lowest non-trivial order in the coupling constant) that at their stable infrared fixed points these operators measure quantum monodromies, in agreement with previous semiclassical studies. Our results help clarify the general relationship between boundary transfer matrices and defect lines, which parallels the relation between (non-commutative) fields on (a stack of) D-branes and their push-forwards to the target-space bulk.

Figures (2)

• Figure 1: A push-forward of the D-brane fields to the target manifold, makes it possible to push the integration contour of (\ref{['defn']}) from the boundary to the interior of the worldsheet.
• Figure 2: The RG flow diagramfor the general symmetric defects discussed in the text. The space of couplings is parametrised by$(a, b) =-\kappa (\lambda , \bar{\lambda})$. The stable fixed points correspond to chiral defects, but there is also an unstable fixed point along the line $\lambda =\bar{\lambda}$ .