Confluent primary fields in the conformal field theory

For any complex simple Lie algebra, we generalize primary fileds in the Wess-Zumino-Novikov-Witten conformal field theory with respect to the case of irregular singularities and we construct integral representations of hypergeometric functions of confluent type, as expectation values of products of generalized primary fields. In the case of sl(2), these integral representations coincide with solutions to confluent KZ equations. Computing the operator product expansion of the energy-momentum tensor and the generalized primary field, new differential operators appear in the result. In the case of sl(2), these differential operators are the same as those of the confluent KZ equations.Comment: 15 pages. Corrected typos. Proposition 3.1 rewritten. Other minor changes, title change

Quantum Painlev\'e Equations: from Continuous to Discrete

We examine quantum extensions of the continuous Painlev\'e equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlev\'e equations II, IV and V. From their auto-B\"acklund transformations we derive the contiguity relations which we interpret as the quantum analogues of the discrete Painlev\'e equations.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Irregular conformal blocks, with an application to the fifth and fourth Painlev\'e equations

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 Painlev\'e equations, using expansions of irregular conformal blocks at an irregular singular point.Comment: 26 page