A note on half-integer irregular representations of Virasoro algebra

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 $(A_1, A_{\text{even}})$ and $(A_1, D_{\text{odd}})$. 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 $\mathbb{Z}_2$-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 $3/2$ and $5/2$. Our analysis clarifies the structure of half-integer irregular modules and provides tools for computing the corresponding irregular states relevant for Argyres-Douglas theories.