Algebraic and combinatorial analysis of elliptic lattice models

Research Project, 2011 – 2013

We will study mathematics related to elliptic lattice models of statistical mechanics, such as the eight-vertex model, the eight-vertex-solid-on-solid (8VSOS) model and the three-colour model. Part of the physical interest in these models comes from their close relation to quantum spin chains and to conformal field theory. Quantum groups are deformed Lie groups, which arose from attempts to formalize the solution of models like those mentioned above. We will develop harmonic analysis on Felder´s quantum group, which is constructed from the 8VSOS model. This will be used to construct new solutions to the quantum Knizhnik-Zamolodzhikov-Bernard equation, which is of importance in conformal field theory. Elliptic hypergeometric functions is a new type of special functions, which appeared from the study of the 8VSOS model. We will prove a number of results on multivariable elliptic hypergeometric functions. Such functions have recently found applications to supersymmetric field theory, and there are also applications in combinatorics. The three-colour model can be obtained as a special case of the 8VSOS model. We will investigate relations between the three-colour model, affine Lie algebras and Painleve equations. We also plan to study macroscopic boundary effects for the three-colour model, and to generalize the recently proved Razumov-Stroganov conjecture to this context.

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