Liouville irregular states of half-integer ranks

A bstract 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 even ) and ( A 1 , D 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 2 n ) to ( A 1 , D 2 n− 1 ). For rank 3 / 2, we solve the conjectured differential equations and find a power series expression for the irregular state | I (3/2) 〉. For rank 5 / 2, our conjecture is consistent with the differential equations recently discovered by H. Poghosyan and R. Poghossian.