Abstract
We investigate the structure of certain protected operator algebras that arise in three-dimensional {mathcal{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 {mathcal{N}=4} SCFTs.
Highlights
An important feature of conformal field theories — and one that has received increased attention in recent years — is that they admit a nonperturbative algebraic formulation in terms of the operator product expansion (OPE) of local operators
The basic data is taken to be the collection of local operators — organized into representations of the conformal algebra and any additional symmetry algebras — and the OPE coefficients that serve as structure constants for the operator algebra
The project of extracting useful results from crossing symmetry is known as the conformal bootstrap
Summary
An important feature of conformal field theories — and one that has received increased attention in recent years — is that they admit a nonperturbative algebraic formulation in terms of the operator product expansion (OPE) of local operators. Aside from being a surprising connection between two- and four- (or six-)dimensional physics, the appearance of chiral algebras in the context of higher dimensional SCFTs is exciting because chiral algebras are strongly and tractably constrained by associativity This means that a wealth of CFT data can potentially be recovered from limited input by solving the chiral algebra bootstrap problem for the theory in question, see for example [13,14]. Deformation quantization of hyperkähler cones has been a subject of considerable study by mathematicians, in the context of geometric representation theory [28,29].3 At first blush, this appears to be a major boon, since there is a classification theorem for these algebras that suggests a finite-dimensional space of solutions to our bootstrap problem. In two appendices we provide background about the algebraic geometry of hyperkähler cones and details of the calculations of moment map two-point functions using supersymmetric localization
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