Liouville Irregular States of Half-Integer Ranks

Abstract: We conjecture a set of differential equations that characterizes the Liouville irregular states of half-integer ranks, which extends the generalized AGT correspondence to all the $(A_1,A_\text{even})$ and $(A_1,D_\text{odd})$ types Argyres-Douglas theories. For lower half-integer ranks, our conjecture is verified by deriving it as a suitable limit of a similar set of differential equations for integer ranks. This limit is interpreted as the 2D counterpart of a 4D RG-flow from $(A_1,D_{2n})$ to $(A_1,D_{2n-1})$. For rank $3/2$, we solve the conjectured differential equations and find a power series expression for the irregular state $|I^{(3/2)}\rangle$. For rank $5/2$, our conjecture is consistent with the differential equations recently discovered by H. Poghosyan and R. Poghossian.