Neveu--Schwarz Irregular Vertex Operators, Decomposition Theorems, and Bilinear Operators

Abstract

We construct rank-zero irregular vertex operators for the Neveu--Schwarz algebra as linear maps between irregular Verma modules of the same rank satisfying the usual superconformal commutation relations and an irregular asymptotic condition. Under the nondegeneracy assumption $Λ_{2p}\neq0$, we prove their existence and uniqueness. We then extend the decomposition theorem for the Neveu--Schwarz algebra to the irregular setting: the tensor product of the free-fermion Fock module with a rank-$p$ Neveu--Schwarz irregular Verma module decomposes into an infinite direct sum of tensor products of two Virasoro irregular Verma modules. We also prove a compatible decomposition of the irregular vertex operators into tensor products of Virasoro irregular vertex operators. Using suitable pairings and mode insertions, we derive bilinear differential equations for weighted sums of products of Virasoro irregular conformal blocks of types $(0,0,1)$ and $(0,2)$. After explicit parameter identifications and a gauge transformation in the Painlevé V case, the resulting bilinear differential operators agree with those appearing in the quantum Painlevé V and IV tau-function equations.