Monodromy of eigenvectors for trigonometric Gaudin algebras

Abstract: Consider a tensor product of representations of a semisimple Lie algebra g. The Gaudin algebra is a commutative algebra which acts on this tensor product, commuting with the action of g. This algebra depends on a parameter which lives in the moduli space of marked genus 0 curves. In previous work, we studied the monodromy of eigenvectors for this algebra as the parameter varies in the real locus of this space. In new work in-progress, we consider trigonometric Gaudin algebras, which act on the same vector space (but do not commute with the g-action). We see that this leads to the action of the affine cactus group, and we describe the action of this group combinatorially using crystals. I will also describe the (conjectural) relation between trigonometric Gaudin algebras and the quantum cohomology of affine Grassmannian slices.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

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