Masahiko Miyamoto
University of Tsukuba

An introduction to a new stage of vertex operator algebra theory

Many functions of the form equal to a (convergent) infinite linear combination of powers of q, with the powers of q of the form n+r where n is a nonnegative integer and r is a fixed rational number, have been discovered to be the ``trace functions'' of certain interesting graded vector spaces, and such functions have been studied from different points of view. For example, many modular functions, including the modular function J, have been found to be the trace functions of vertex operator algebras. This is intimately related to ``monstrous moonshine'' and the ``moonshine module vertex operator algebra.'' I will explain such ideas, with motivation, including a sketch of the concept of vertex operator algebra. Then I would like to introduce a new kind of trace function and show that functions of the more general form in which the powers of q are multiplied by finitely many nonnegative integral powers of ln q, like solutions of differential equations of regular singular type, can also be given as the trace functions of vertex operator algebras. A classical result on symmetric (Frobenius) algebras enters here. I will discuss consequences of this result.