12:00 pm Wednesday, January 15, 2014
Special joint meeting of Geometry and Geometry and Strings : Siebert-Witten curves and double bruhat cells by Harold Williams (Berkeley) in 8.136
Seiberg-Witten Curves and Double Bruhat Cells Take your favorite finite-type Dynkin diagram, equip it with a sink-source orientation, and take its (square) product with a Kronecker quiver: for SU(N) the result should be a quiver that looks something like N-1 squares glued together side-by-side, with all the vertical arrows doubled. In the last decade this quiver has turned out to lead a double life in mathematics and physics: physically it encodes part of the BPS spectrum of pure N=2 Yang-Mills theory, or the intersection numbers of a certain homology basis of the corresponding Seiberg-Witten curves. Mathematically it describes the cluster coordinates on a special double Bruhat cell of the corresponding Lie group (hence encodes its Poisson and totally positive structure) which in particular is the phase space of an open Toda system. In either setting, via the formalism of cluster transformations we can read off a distinguished rational automorphism from this quiver, which is on one hand a sort of noncommutative generating function of BPS indices, and on the other hand a symmetry of the Toda system (in fact a discrete recurrence called the Q-system, arising in the representation theory of quantum loop algebras). The goal of the talk is to convince you that there's actually a good reason for the same quiver to appear in these seemingly unrelated contexts, with certain irregular Hitchin systems and the spectral networks of Gaiotto, Moore, and Neitzke playing the key intermediate rol
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