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Hypergeometric integrals in integrable systems

Andrew Kels
Wednesday, December 19, 2018 - 14:00

In this talk I will consider connections between two different types of integrable systems.  The first type are 2-dimensional lattice models of statistical mechanics, which are considered integrable if they satisfy a Yang-Baxter equation.  The simplest and most-well known example of this type is the 2-dimensional Ising model. The second type are discrete soliton equations, with a well-known example being the discrete (lattice potential) KdV equation.  For the latter type of equations, the property of "consistency-around-a-cube" (or 3D-consistency) has been proposed as a condition of integrability, since this property immediately implies a Lax pair and Backlund transformation for the respective equations.  An important classification of such 3D-consistent equations (with some extra assumptions) was given by Adler, Bobenko, and Suris (ABS), resulting in a list of 7 main equations.In this talk I will show how all integrable 3D-consistent equations in the ABS classification, arise from an asymptotic limit of a counterpart hypergeometric integral. Arguably the simplest example is the lattice potential KdV equation, which may be derived from an asymptotic limit of the Euler beta function.  Furthermore, each of the respective hypergeometric integrals, has an interpretation as a Yang-Baxter equation for an integrable lattice model of statistical mechanics.  This provides a direct (quantum/classical) correspondence between two types of integrable equations; the Yang-Baxter equation, and a 3D-consistent quadrilateral equation. Finally, I will discuss how this correspondence extends to the integrable systems themselves, and also present new types of elliptic hypergeometric functions, and related mathematical formulas and applications, which arose from the above considerations (along with connections to supersymmetic gauge theory).

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