Masahiko Miyamoto
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.