205 research outputs found
Hypergeometric solutions to Schr\"odinger equations for the quantum Painlev\'e equations
We consider Schr\"odinger equations for the quantum Painlev\'e equations. We
present hypergeometric solutions of the Schr\"odinger equations for the quantum
Painlev\'e equations, as particular solutions. We also give a representation
theoretic correspondence between Hamiltonians of the Schr\"odinger equations
for the quantum Painlev\'e equations and those of the KZ equation or the
confluent KZ equations.Comment: 17 pages; Journal of Mathematical Physics (Vol.52, Issue 8) 201
CFT approach to the -Painlev\'e VI equation
Iorgov, Lisovyy, and Teschner established a connection between isomonodromic
deformation of linear differential equations and Liouville conformal field
theory at . In this paper we present a analog of their construction.
We show that the general solution of the -Painlev\'e VI equation is a ratio
of four tau functions, each of which is given by a combinatorial series arising
in the AGT correspondence. We also propose conjectural bilinear equations for
the tau functions.Comment: 26 page
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
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
Quantum Painlevé Equations: from Continuous to Discrete
We examine quantum extensions of the continuous Painlevé equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlevé equations II, IV and V. From their auto-Bäcklund transformations we derive the contiguity relations which we interpret as the quantum analogues of the discrete Painlevé equations
Noncommutative Toda Chains, Hankel Quasideterminants And Painlev'e II Equation
We construct solutions of an infinite Toda system and an analogue of the
Painlev'e II equation over noncommutative differential division rings in terms
of quasideterminants of Hankel matrices.Comment: 16 pp; final revised version, will appear in J.Phys. A, minor changes
(typos corrected following the Referee's List, aknowledgements and a new
reference added
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