Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks
Giovanni Felder, Gabriele Rembado
TL;DR
This work develops a representation-theoretic framework for irregular WZNW conformal blocks by introducing singular modules for affine Lie algebras at noncritical level, parameterised by tame and wild data that mirror irregular meromorphic connections. Using Segal–Sugawara operators, the authors construct spaces of irregular covacua (coinvariants) that assemble into flat bundles over spaces of tame isomonodromy times, recovering the KZ, dynamical KZ, and irregular isomonodromy structures in various limits. The paper provides PBW bases, gradings, and filtrations for both affine and finite-depth modules, establishes finite-dimensionality results for coinvariants in the tame setting, and develops universal flat connections on tensor powers of truncated current algebras that quantise irregular isomonodromic systems. It also connects to broader themes such as wild character varieties, Takiff algebras, and quantum isomonodromy, outlining future directions toward irregular generalisations and higher-genus extensions. Overall, the authors offer a cohesive algebraic construction of irregular conformal blocks and their flat connections, tying together affine representation theory, isomonodromic deformation theory, and irregular CFT phenomena.
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 à la Klarès, 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.