Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks

TL;DR

This paper defines irregular vacua and conformal blocks for affine Lie algebras at generic levels, establishing their geometric and algebraic structures, and introduces a universal flat connection generalizing known irregular KZ connections.

Contribution

It provides a rigorous mathematical framework for irregular conformal blocks and constructs a universal flat connection extending previous irregular KZ connections.

Findings

Spaces of irregular conformal blocks form flat vector bundles.

A universal flat connection generalizing irregular KZ is constructed.

The approach links affine Lie algebra modules with moduli of irregular connections.

Abstract

We give a mathematical definition of spaces of irregular vacua/covacua in genus zero, for any simple Lie algebra, working at generic noncritical level. This uses coinvariants of affine-Lie-algebra modules whose parameters match up with those of moduli spaces of irregular-singular meromorphic connections: the de Rham spaces. The Segal--Sugawara representation of the Virasoro algebra is used to show that the spaces of irregular conformal blocks assemble into a flat vector bundle over the space of somonodromy times \`a la Klar\`es, and we provide a universal version of the resulting flat connection generalising the irregular KZ connection of Reshetikhin and the dynamical KZ connection of Felder--Markov--Tarasov--Varchenko.