Short Star-Products for Filtered Quantizations, I

Abstract

We develop a theory of short star-products for filtered quantizations of graded Poisson algebras, introduced in 2016 by Beem, Peelaers, and Rastelli for algebras of regular functions on hyperKähler cones in the context of 3-dimensional N = 4 superconformal field theories [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392]. This appears to be a new structure in representation theory, which is an algebraic incarnation of the non-holomorphic SU(2)-symmetry of such cones. Using the technique of twisted traces on quantizations (an idea due to Kontsevich), we prove the conjecture by Beem, Peelaers, and Rastelli that short star-products depend on finitely many parameters (under a natural nondegeneracy condition), and also construct these star products in a number of examples, confirming another conjecture by Beem, Peelaers, and Rastelli.