Vertex Operator Algebra - Vertex Operator Superalgebra

Vertex Operator Superalgebra

When the underlying vector space V has a Z₂ grading, so that it splits as a sum of even and odd parts

with 1 in V₊, the structure of a vertex superalgebra can be defined on V by incorporating the usual rule of signs in the axiom for the four point function:

• (Four point function) For any a, b, c ∈ V_±, there is an element

such that Y(a,z)Y(b,w)c, εY(b,w)Y(a,z)c, and Y(Y(a,z-w)b,w)c are the expansions of X(a,b,c;z,w) in V((z))((w)), V((w))((z)), and V((w))((z-w)), respectively, where ε is -1 if both a and b are odd and 1 otherwise.

If in addition there is a Virasoro element ω in the even part of V₂, then V is called a vertex operator superalgebra. One of the simplest examples is the vertex operator superalgebra generated by a single complex fermion.