TITLE:
Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces
AUTHORS:
Alexander Zuevsky
KEYWORDS:
Vertex Operator Superalgebras; Intertwining Operators; Riemann Surfaces; Szegö Kernel; Modular Forms; Theta-Functions; Frobenius—Fay and Jacobi Product Identities
JOURNAL NAME:
Applied Mathematics,
Vol.4 No.10C,
October
8,
2013
ABSTRACT: We review our recent results on computation of the
higher genus characters for vertex operator superalgebras modules. The vertex
operator formal parameters are associated to local parameters on Riemann
surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus
Riemann surfaces. For the free fermion vertex operator superalgebra we present
a closed formula for the genus two continuous orbifold partition functions (in
either sewings) in terms of an infinite dimensional determinant with entries
arising from the original torus Szeg? kernel. This partition function is
holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical
(including Fay’s and Jacobi triple product) identities show up in a natural way
in the vertex operator algebra approach.