A note on half-integer irregular representations of Virasoro algebra

Yichi Zang

Paper

A note on half-integer irregular representations of Virasoro algebra

Abstract

We study irregular representations of Virasoro algebra associated with half-integer order singularities, which arise naturally in the 2d CFT description of Argyres-Douglas theories of type

and

. While integer-rank irregular states admit a well-established free-field construction, the half-integer case is more subtle due to the presence of branch cuts. In this note, we present two equivalent constructions of half-integer irregular representations. The first one is based on a

-twisted free boson, which is motivated from the monodromy structure of Hitchin system. The second one employs a recursion relation of the Virasoro eigenvalues recently proposed in the literature. We explicitly demonstrate the equivalence of these two parameterization schemes at rank

and

. Our analysis clarifies the structure of half-integer irregular modules and provides tools for computing the corresponding irregular states relevant for Argyres-Douglas theories.