Short Star-Products for Filtered Quantizations, I

TL;DR

This paper introduces short star-products for filtered quantizations of graded Poisson algebras, confirming their finite-parametric nature and providing explicit constructions, thus revealing a new structure linked to non-holomorphic SU(2) symmetry in hyperKähler cones.

Contribution

It develops a theory of short star-products for filtered quantizations, proves their finite-parameter dependence, and constructs explicit examples, advancing the understanding of algebraic structures in representation theory.

Findings

Short star-products depend on finitely many parameters.

Constructed explicit examples of short star-products.

Confirmed conjectures related to their structure and parametrization.

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 hyperK\"ahler 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, arXiv:1601.05378]. This appears to be a new structure in representation theory, which is an algebraic incarnation of the non-holomorphic ${\rm SU}(2)$-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.