A supergeometric interpretation of vertex operator superalgebras

Abstract

Conformal field theory (or more specifically, string theory) and related theories (cf. [BPZ], [FS], [V], and [S]) are the most promising attempts at developing a physical theory that combines all fundamental interactions of particles, including gravity. The geometry of this theory extends the use of Feynman diagrams, describing the interactions of point particles whose propagation in time sweeps out a line in space-time, to one-dimensional “particles” (strings) whose propagation in time sweeps out a two-dimensional surface. For genus zero holomorphic conformal field theory, algebraically, these interactions can be described by products of vertex operators or more precisely, by means of vertex operator algebras (cf. [Bo] and [FLM]). However, until 1990 a rigorous mathematical interpretation of the geometry and algebra involved in the “sewing” together of different particle interactions, incorporating the analysis of general analytic coordinates, had not been realized. In [H1] and [H2], motivated by the geometric notions arising in conformal field theory, Huang gives a precise geometric interpretation of the notion of vertex operator algebra by considering the geometric structure consisting of the moduli space of genus zero Riemann surfaces with punctures and local coordinates vanishing at the punctures, modulo conformal equivalence, together with the operation of sewing two such surfaces, defined by cutting discs around one puncture from each sphere and appropriately identifying the boundaries. Important aspects of this geometric structure are the concrete realization of the moduli space in terms of exponentials of a representation of the Virasoro algebra and a precise analysis of sewing using these resulting exponentials. Using this geometric structure, Huang then introduces the notion of geometric vertex operator algebra with central charge c ∈ C, and proves that the category of geometric vertex operator algebras is isomorphic to the category of vertex operator algebras. In [F], Friedan describes the extension of the physical model of conformal field theory to that of superconformal field theory and the notion of a superstring whose propagation in time sweeps out a supersurface. Whereas conformal field theory attempts to describe the interactions of bosons, superconformal field theory attempts to describe the interactions of boson-fermion pairs. This, in particular, requires an operator D such that D 2 = @ @z . Such an operator arises naturally in supergeometry. In [BMS], Beilinson, Manin and Schechtman study some aspects of superconformal symmetry, i.e., the Neveu-Schwarz algebra, from the viewpoint of algebraic geometry. In this work, we will take a differential geometric approach, extending Huang’s geometric interpretation of vertex operator algebras to a supergeometric interpretation of vertex operator superalgebras. Within the framework of supergeometry (cf. [D], [R] and [CR]) and motivated by superconformal field theory, we define the moduli space of super-Riemann surfaces with genus zero “body”, punctures, and local superconformal coordinates vanishing at the punctures, modulo superconformal equivalence. We announce the result that any local superconformal coordinates can be expressed in terms of exponentials of certain superderivations, and that these superderivations give a representation of the Neveu-Schwarz algebra with zero central charge. We define a