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.

💡 Research Summary

This paper investigates the construction of irregular representations of the Virasoro algebra associated with half‑integer order singularities, which appear in the two‑dimensional conformal field theory (CFT) description of certain Argyres‑Douglas (AD) theories, namely the ((A_1,A_{\text{even}})) and ((A_1,D_{\text{odd}})) families. While integer‑rank irregular states have a well‑established free‑field (bosonic) realization, the half‑integer case is more subtle because the underlying Hitchin system exhibits a branch cut, leading to a (\mathbb{Z}_2) monodromy. The author presents two equivalent constructions for these half‑integer irregular modules and demonstrates their equivalence explicitly at ranks (3/2) and (5/2).

1. Review of integer‑rank irregular states.
For an integer rank (n), the irregular state (|I(n)\rangle) is defined as a coherent state of the modes (\alpha_k) of a free boson (\varphi(z)). The stress‑energy tensor (T(z)= -:\partial\varphi,\partial\varphi:+ Q\partial^2\varphi) yields Virasoro generators (L_k) that act on (|I(n)\rangle) as \