Formal differential operators, vertex operator algebras and zeta-values, II

Abstract

We introduce certain correlation functions (graded q-traces) associated to vertex operator algebras and superalgebras which we refer to as n-point functions. These naturally arise in the studies of representations of Lie algebras of differential operators on the circle (J. Lepowsky, to appear, J. Lepowsky, In: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Amer. Math. Soc., Providence, RI, 1999, p. 327, A. Milas, Formal Differential Operators, Vertex Operator Algebras and Zeta-values, I, to appear). We investigate their properties and consider the corresponding graded q-traces in parallel with the passage from genus 0 to genus 1 conformal field theory. By using the vertex operator algebra theory we analyze in detail correlation functions in some particular cases. We obtain elliptic transformation properties for q-traces and the corresponding q-difference equations. In particular, our construction leads to certain correlation functions and q-difference equations investigated by Bloch and Okounkov (Adv. Math. 149 (2000), 1).