REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: We will explain a new connection between the theories of cluster algebras and Legendrian knots.  The starting point is the recent construction by Shende-Treumann-Zaslow of a new Legendrian knot invariant. This invariant is a moduli space of objects with various interpretations, for example as sheaves, quiver representations, or objects in a Fukaya category. For specific choices of Legendrian knot, it turns out that these invariants recover essentially all examples of cluster varieties related to the group SL_n.  Cluster varieties are spaces that have been realized to play a wide range of roles in geometry and representation theory in the past decade, and include character varieties of punctured surfaces and strata of algebraic groups and homogeneous spaces.  Our main result is that on such a variety the cluster structure, which can be regarded as an atlas of open toric subsets with special properties and transition functions, is in a sense determined by the structure of the set of exact Lagrangian fillings of the associated Legendrian knot. This is joint work in progress with Vivek Shende, David Treumann, and Eric Zaslow.