Rank-zero irregular Neveu-Schwarz vertex operators are shown to exist uniquely, and their decomposition into Virasoro irregular vertex operators yields bilinear operators that coincide with the quantum Painlevé V and IV tau-function equations.
Degeneration limits of Virasoro vertex operators and Painlev\'e tau functions
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abstract
We construct degeneration limits of vertex operators for the Virasoro algebra. Our method relies on the rearranged expansion of compositions of vertex operators together with their integral representations. Using this framework, we obtain a vertex operator between Verma modules of rank $r+1$ as a degeneration of a composition of two vertex operators between Verma modules of rank $r$ ($r\in\mathbb{Z}_{\geq 0}$). Furthermore, we apply these degeneration limits to prove the conjectural expansions of the $\tau$ functions of the fifth and fourth Painlev\'e equations in terms of irregular conformal blocks [H. Nagoya, J. Math. Phys. 56, 123505 (2015)].
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Neveu--Schwarz Irregular Vertex Operators, Decomposition Theorems, and Bilinear Operators
Rank-zero irregular Neveu-Schwarz vertex operators are shown to exist uniquely, and their decomposition into Virasoro irregular vertex operators yields bilinear operators that coincide with the quantum Painlevé V and IV tau-function equations.