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Title:
Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs
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Abstract:
The star-triangle relation is a distinguished form of the quantum Yang-Baxter equation which plays a fundamental role in the theory of two-dimensional integrable models of statistical mechanics and quantum field theory. In this talk I will explain a connection between the star-triangle relation and classical integrable evolution equations on quad-graphs (planar graphs with quadrilateral faces). In particular, I will show how to use a special "master solution" of the star-triangle relation to reproduce the Adler-Bobenko-Suris (ABS) classification of classical integrable equations, originally obtained through the consistency-around-a-cube approach. As an example I will consider a lattice model which describes quantum fluctuations of circle patterns and the associated discrete conformal transformations connected with the Thurston’s discrete analogue of the Riemann mappings theorem. In the quasi-classical limit the model precisely describe the geometry of integrable circle patterns with prescribed intersection angles on the (hyperbolic) plane.
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