Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations

Abstract

We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators. Our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points. As an application, we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth Painlevé equations, using expansions of irregular conformal blocks at an irregular singular point.

26 pages

References in corpus (6)

• Isomonodromic tau-functions from Liouville conformal blocks
• Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial
• Bilinear equations on Painleve tau functions from CFT
• Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks
• Virasoro irregular conformal block and beta deformed random matrix model
• Confluent primary fields in the conformal field theory

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