Poisson brackets and the string Lie algebra of a Calabi-Yau category

Abstract: Goldman defined a symplectic structure on the moduli space of local systems on a closed oriented surface, constructed a collection of natural Hamiltonians on the moduli space by taking trace of monodromy around loops on the surface, and computed Poisson brackets among these Hamiltonians in terms of what is now called the Goldman bracket on free homotopy classes of loops. Chas and Sullivan generalized the Goldman bracket to a string bracket on the degree-shifted equivariant homology of the free loop space of a closed oriented manifold of any dimension, but the compatibility with the corresponding shifted-symplectic geometry on the moduli space of local systems remained mostly conjectural. We generalize these results of Goldman and of Chas-Sullivan to higher dimensional ’non-commutative’ closed oriented manifolds in the form of smooth Calabi-Yau categories. Our main results are the description of a chain-level ’string Lie bracket’ on cyclic chains of a smooth Calabi-Yau category and the intertwining of this string Lie bracket on cyclic chains with the shifted Poisson bracket on functions on the moduli space of objects in the category.

mathematical physicscommutative algebraalgebraic geometryrings and algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

( slides )

Comments: Meeting Link

uni-koeln.zoom.us/j/91875528987?pwd=TjNZV1lNdVJNOWUrR2xNalRuQWk3dz09

Meeting ID: 918 7552 8987 Password: LAGOON2022

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Longitudinal Algebra and Geometry Open ONline Seminar (LAGOON)

Series comments: Description: Research webinar series

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