Communications in Number Theory and Physics

Volume 13 (2019)

Number 1

Approximating tau-functions by theta-functions

Pages: 203 – 223



B. Dubrovin (Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy; and the N.N. Bogolyubov Laboratory of Geometrical Methods in Mathematical Physics, Moscow State University, Moscow, Russia)


We prove that the logarithm of an arbitrary tau-function of the 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, we consider theta-functional approximations of the Witten–Kontsevich tau-function.


The work is supported by the Russian Science Foundation Grant No. 16-11-10260 “Geometry and Mathematical Physics of Integrable Systems”.

2010 Mathematics Subject Classification

Primary 37K10. Secondary 14K25.

Received 18 July 2018

Accepted 24 October 2018

Published 29 April 2019