the bulk, so that it depends on left-moving currents only. As a result, it commutes with the right-moving algebra along the entire RG flow. At the infrared fixed point, this bulk operator also commutes with the left-moving algebra, and can be identified with the topological Wilson loops of ref. [@Felder:1999cv], or with the topological defect lines of ref. [@Petkova:2000ip]. The fixed-point operator is in fact the trace of the quantum monodromy matrix \[37–40\]. Though we do not have a full proof, we will verify these statements explicitly at low orders in the perturbative expansion.

The existence of renormalisable loop operators, which can be moved freely to the boundary from the bulk, is a remarkable feature of the WZW model. It implies universal RG flows, common to all ultraviolet fixed points. From the geometric point of view, it means that the corresponding D-brane fields admit special, universal push-forwards to the entire target manifold. It would be very interesting to understand whether and, if so, how these features could be generalised to other models.

Besides those cited above, several other papers touch upon various aspects of our work. Most closely related is ref.[@Hikida:2001py], whose authors studied the same transfer-matrix operator and its 