Vertex operator superalgebras

Abstract

In this paper, we construct the action of super vertex operators In section 1, we construct a Fock space V for a 2l-dimensional C-vector space A. In physics, V appears as the Fock space of Neveu-Schwarz model of dual poins. See Neveu-Schwarz [5]. In mathematics, V appears as the spin representation space of the affine Lie algebra&ŝ2l. See Frenkel [2] and [3]. In section 2, we define a super vertex operator A V,Z : V→V[[z,z-1]] for each element v of the Fock space V. The most important property of the super vertex operators is the so-called associative law. Using this law, we can see that the coefficients v n : V→V in A v,z span a Lie superalgebra and explicitly write down the Lie superbackets in section 3. It is a huge Lie superalgebra called the vertex operator superalgebra of A.