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 ([BPR]). 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 of [BPR] 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 of [BPR].

a detailed study of twisted traces on quantizations of Kleinian singularities of type A n−1 . In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed by Beem, Peelaers and Rastelli in connection with 3-dimensional superconformal field theory. In particular, we confirm their conjecture that for n ≤ 4 a unitary short star-product is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers and Rastelli. If n = 2, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for sl 2 . Thus the results of this paper may be viewed as a starting point for a generalization of the theory of unitary Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short star-products corresponding to twisted traces, which are generalizations of discrete Painlevé systems.

We give a new proof of a recent result of Munteanu–Wang relating scalar curvature to volume growth on a 3 3 -manifold with non-negative Ricci curvature. Our proof relies on the theory of μ \mu -bubbles introduced by Gromov [Geom. Funct. Anal. 28 (2018), pp. 645–726] as well as the almost splitting theorem due to Cheeger–Colding [Ann. of Math. (2) 144 (1996), pp. 189–237].