Hochschild and Poisson (co)homology: finiteness and formality
Travis Schedler (MPIM & London)
Abstract: The Hamiltonian approach views quantum systems as noncommutative formal deformations of classical ones; the first-order part of the deformation is a symplectic or Poisson structure on the classical phase space. Kontsevich’s formality theorem famously explains how the deformation theory of the classical and quantum systems coincide (the titular Poisson and Hochschild cohomology). In cases of smooth symplectic varieties, this is controlled by de Rham cohomology, which is finite-dimensional. I will survey how these finite-dimensionality and formality properties extend, or do not extend, to more general (singular, degenerate, or orbifold) situations, and how to approach these problems using D-modules on the the classical phase space. This includes joint works with Etingof, Negron, and Pym.