Twisted Vertex Algebra Modules for Irregular Connections: A Case Study

A vertex algebra with an action of a group $G$ comes with a notion of $g$-twisted modules, forming a $G$-crossed braided tensor category. For a Lie group $G$, one might instead wish for a notion of $(\textrm{d}+A)$-twisted modules for any $\mathfrak{g}$-connection on the formal punctured disk. For connections with a regular singularity, this reduces to $g$-twisted modules, where $g$ is the monodromy around the puncture. The case of an irregular singularity is much richer and involved, and we are not aware that it has appeared in vertex algebra language. The present article is intended to spark such a treatment, by providing a list of expectations and an explicit worked-through example with interesting applications. Concretely, we consider the vertex super algebra of symplectic fermions through its associated Clifford algebra and study its twisted module with respect to irregular $\mathfrak{s}\mathfrak{l}_{2}$-connections. We first determine the category of representations, depending on the formal type of the connection. Then we prove that a Sugawara-type construction gives a Virasoro action and we prove that as Virasoro modules our representations are direct sums of Whittaker modules. Conformal field theory with irregular singularities resp. wild ramification appear in the context of geometric Langlands correspondence, in particular work by Witten [51], and more generally in higher-dimensional context. Our original motivation in [24] comes from semiclassical limits of the generalized quantum Langlands kernel, which fibres over the space of connections (similar to the affine Lie algebra at critical level) and as such gives a family of deformations of the Feigin–Tipunin algebra. Our present article now describes, in the smallest case, the fibres and their representation categories over irregular connections.