ERG Kernels on Multiply Connected Configuration Spaces

In the functional-integral formulation of Euclidean field theory, exact renormalization group (ERG) transformations are realized by functional-integral kernels.

Unlike the ERG flow equations that describe infinitesimal ERG transformations, these ERG kernels explicitly depend on the global topology of the configuration space.

This paper explores this topology dependence for multiply connected configuration spaces.

We show that the ERG kernel is in general given by a weighted sum of kernels on its universal covering space, where the weight factors are determined by a one-dimensional representation of the fundamental group.

These weight factors are shown never to be renormalized under the ERG.

We also show that these factors can be interpreted as Aharonov-Bohm phases with respect to a background magnetic flux penetrating the infinite-dimensional configuration space.

From this viewpoint, a normalization condition for ERG transformations corresponds to a flux-quantization condition, which is equivalent to the level-quantization condition for Wess-Zumino-Witten terms in nonlinear sigma models.

Finally, we present an alternative gauge-equivalent form of the ERG flow equation that incorporates this topological information locally.