Short star-products for filtered quantizations

Seminar: 
Geometry, Symmetry and Physics
Event time: 
Monday, October 7, 2019 - 4:30pm
Location: 
LOM 214
Speaker: 
Pavel Etingof
Speaker affiliation: 
MIT
Event description: 

Abstract: Let A be a filtered Poisson algebra with Poisson bracket {, } of degree −2. A star product on A is an associative product ∗ : A ⊗ A → A given by a ∗ b = ab + X i≥1 Ci(a, b), where Ci has degree −2i and C1(a, b) − C1(b, a) = {a, b}. We call the product * even if Ci(a, b) = (−1)iCi(b, a) for all i, and call it short if Ci(a, b) = 0 whenever i > min(deg(a), deg(b)). Motivated by three-dimensional N = 4 superconformal field theory, In 2016 Beem, Peelaers and Rastelli considered short even starproducts for homogeneous symplectic singularities (more precisely, hyperK¨ahler cones) and conjectured that that they exist and depend on finitely many parameters. We prove the dependence on finitely many parameters in general and existence for a large class of examples, using the connection of this problem with zeroth Hochschild homology of quantizations suggested by Kontsevich. Beem, Peelaers and Rastelli also computed the first few terms of short quantizations for Kleinian singularities of type A, which were later computed to all orders by Dedushenko, Pufu and Yacoby. We will discuss some generalizations of these results. This is joint work with Eric Rains and Douglas Stryker.