April 17, 2023 to July 14, 2023
Institut Henri Poincaré, Paris
Europe/Paris timezone

Ezra Getzler : Symplectic forms on derived stacks

Jul 3, 2023, 10:00 AM


Abstract: Chern-Weil theory gives an explicit formula for a differential form on BGL(n) representing the Chern character. The lowest components of this form are of importance in geometry: the component of total degree 2 is the first Chern class, while the component of total degree 4 component is the shifted symplectic form on BGL(n). This formula was first obtained in the Berkeley thesis of Shulman in 1972. The extension to the derived stack of perfect complexes (essentially the generalization to graded vector spaces of BGL(n)) is more difficult: an existence proof was obtained by Toën and Vezzossi, but their approach does not lead to a formula. In this talk, I show how, using an explicit realization of this derived stack (joint work with Kai Behrend) and negative cyclic homology, we obtain an explicit formula for a differential form representing the Chern character on this derived stack, and hence also an explicit formula for the shifted symplectic form.

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