Deformation quantization and superconformal symmetry in three dimensions

TL;DR

This work identifies and analyzes protected associative algebras in 3d N=4 SCFTs, showing they arise as deformation quantizations of Higgs- or Coulomb-branch chiral rings with a natural, canonical basis that aligns quantum structure constants with CFT OPE data. It develops a robust bootstrap framework incorporating equivariance, selection rules, and unitarity to constrain the star-product structure, and connects these algebras to deformation quantization of hyperkähler cones, classified by geometric data such as periods modulo Namikawa Weyl symmetries. The authors illustrate the program with explicit examples: minimal nilpotent orbits yield higher-spin algebras like hs[lambda], while Kleinian A-type singularities require navigating gauge-fixing and unitarity to isolate allowed canonical bases, often matching known gauge-theory constructions. The results suggest a finite, gauge-fixed space of canonical protected algebras for a given moduli-space geometry, with potential ties to localization, omega-deformation, and broader geometric representation theory.

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.

Figures (6)

• Figure 1: Regions in the $(\kappa, \alpha)$ plane that are not excluded by the constraints of unitarity for operators of dimension less than or equal to two (top left), three (top right), four (bottom left), and five (bottom right).
• Figure 2: Allowed values of $(\kappa,\alpha)$ after imposing unitarity bounds for the operators listed in the text. The horizontal dotted black line is at $\alpha = (18-\frac{144}{\pi^2})/72$, which is the value of $\alpha$ in the $A_2$ affine quiver gauge theory. The black dot corresponds to the $\mathbb{Z}_3$ gauging of a free hypermultiplet.
• Figure 3: Close up views of the unitarity bounds for coefficients $\kappa$ and $\alpha$ near the points of interest for the free (left) and affine quiver (right) theories. The allowed values lie in the shaded region. The black dot in the left plot indicates the location of the $\mathbb{Z}_3$ gauge theory of the free hypermultiplet. The dotted black line denotes value of $\alpha$ for the affine $A_2$ quiver gauge theory.
• Figure 4: Constraints imposed by unitarity on the parameters of the $A_3$ star product. The allowed regions satisfy unitarity bounds for operators of dimension up to two (upper left), three (upper right), and four (lower left). In the lower right we show at higher resolution the intersection of the allowed region with the plane representing the value of $\alpha$ relevant to the affine $A_3$ quiver gauge theory.
• Figure 5: Allowed values of the period parameters for the $A_3$ theory when $\alpha$ is set equal to its value for the $A_3$ affine quiver gauge theory. The bounds correspond to requiring the correct sign for two-point functions of operators up to dimension two (top left), three (top right), four (bottom left), and five (bottom right).

Theorems & Definitions (1)

• Conjecture 1