Deformation quantization and superconformal symmetry in three dimensions

TL;DR

This paper explores the structure of protected operator algebras in 3D N=4 superconformal field theories, revealing their interpretation as quantizations of BPS chiral rings and identifying unique bases related to higher spin algebras and symplectic reflection algebras.

Contribution

It introduces a framework for understanding these operator algebras as quantizations with canonical bases, providing new insights into their structure and uniqueness in certain cases.

Findings

Quantum algebras correspond to quantized chiral rings.

Unique canonical bases exist for minimal nilpotent orbit cases.

Evidence suggests at most one canonical basis per quantum algebra.

Abstract

We investigate the structure of certain protected operator algebras that arise in three-dimensional N=4 superconformal field theories. We find that these algebras can be understood as a quantization of (either of) the half-BPS chiral ring(s). An important feature of this quantization is that it has a preferred basis in which the structure constants of the quantum algebra are equal to the OPE coefficients of the underlying superconformal theory. We identify several nontrivial conditions that the quantum algebra must satisfy in this basis. We consider examples of theories for which the moduli space of vacua is either the minimal nilpotent orbit of a simple Lie algebra or a Kleinian singularity. For minimal nilpotent orbits, the quantum algebras (and their preferred bases) can be uniquely determined. These algebras are related to higher spin algebras. For Kleinian singularities the algebras can be characterized abstractly - they are spherical subalgebras of symplectic reflection algebras - but the preferred basis is not easily determined. We find evidence in these examples that for a given choice of quantum algebra (defined up to a certain gauge equivalence), there is at most one choice of canonical basis. We conjecture that this is the case for general N=4 SCFTs.