In this talk I will consider connections between two different types of integrable systems. The first type are 2dimensional lattice models of statistical mechanics, which are considered integrable if they satisfy a YangBaxter equation. The simplest and mostwell known example of this type is the 2dimensional Ising model. The second type are discrete soliton equations, with a wellknown example being the discrete (lattice potential) KdV equation. For the latter type of equations, the property of "consistencyaroundacube" (or 3Dconsistency) 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 3Dconsistent 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 3Dconsistent 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 YangBaxter equation for an integrable lattice model of statistical mechanics. This provides a direct (quantum/classical) correspondence between two types of integrable equations; the YangBaxter equation, and a 3Dconsistent 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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Hypergeometric integrals in integrable systems
Research Group:
Andrew Kels
Institution:
SISSA
Location:
A137
Schedule:
Wednesday, December 19, 2018  14:00
Abstract:
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