Approximating tau-functions by theta-functions (Q2418750)

The author proves that the logarithm of an arbitrary tau-function of the Korteweg-de Vries (KdV) hierarchy can be approximated, in the topology of graded formal series, by the logarithmic expansions of hyperelliptic theta functions of finite genus, up to at most quadratic terms. As an example, he considers theta-functional approximations of the Witten-Kontsevich tau-function. The paper is organized as follows. The first section is an introduction to the subject. In the second section the author gives the proofs of the results. The third section is devoted to some examples: logarithmic expansions of hyperelliptic theta functions and theta functional approximations of the Witten-Kontsevich tau-function. The constructions of the paper can be generalized to other spaces of matrix-valued series. Moreover they can be extended to series with coefficients in an arbitrary simple Lie algebra.