Fusion Residues

TL;DR

This paper explores the conditions under which Verlinde dimensions of rational conformal field theories can be represented as correlation functions in topological Landau-Ginzburg theories, highlighting the role of fusion rule symmetries and specific perturbations.

Contribution

It demonstrates that certain perturbations of Grassmannian superpotentials relate Verlinde dimensions to topological LG residues, proposing a connection to Grassmannian topological sigma models.

Findings

Topological LG residues can represent Verlinde dimensions under specific conditions.

Perturbations of Grassmannian superpotentials relate to twisted Verlinde dimensions.

Conjecture linking LG correlation functions to Grassmannian sigma models with instanton coupling.

Abstract

We discuss when and how the Verlinde dimensions of a rational conformal field theory can be expressed as correlation functions in a topological LG theory. It is seen that a necessary condition is that the RCFT fusion rules must exhibit an extra symmetry. We consider two particular perturbations of the Grassmannian superpotentials. The topological LG residues in one perturbation, introduced by Gepner, are shown to be a twisted version of the $SU(N)_k$ Verlinde dimensions. The residues in the other perturbation are the twisted Verlinde dimensions of another RCFT; these topological LG correlation functions are conjectured to be the correlation functions of the corresponding Grassmannian topological sigma model with a coupling in the action to instanton number.