A supergeometric interpretation of vertex operator superalgebras

Abstract

Huang's geometric interpretation of vertex operator algebras is extended to a supergeometric interpretation of vertex operator superalgebras. In particular, the geometry of spheres with punctures and local analytic coordinates in terms of exponentials of derivations is extended to the geometry of superspheres with punctures, a given spin structure, and local superconformal coordinates in terms of exponentials of superderivations. The notion of supergeometric vertex operator superalgebra is introduced, and in addition, the notion of (superalgebraic) vertex operator superalgebra over a Grassmann algebra and with odd formal variables is introduced. The main result is that the category of supergeometric vertex operator superalgebras over a Grassmann algebra with a given spin structure and the category of vertex operator superalgebras over a Grassmann algebra are isomorphic. In addition, the supergeometric symmetry between the two different possible spin structures naturally gives rise to a symmetry in the superalgebraic structure.