Integrable Systems and Classification of 2-Dimensional Topological Field Theories

Abstract

In this paper we consider the so-called WDVV equations from the point of view of differential geometry and of the theory of integrable systems as defining relations of 2-dimensional topological field theory. A complete classification of massive topological conformal field theories (TCFT) is obtained in terms of the monodromy data of an auxiliary linear operator with rational coefficients. The procedure of coupling a TCFT to topological gravity is described (at tree level) via certain integrable bihamiltonian hierarchies of hydrodynamic type and their τ-functions. A possible role for the bihamiltonian formalism in the calculation of higher genus corrections is discussed. As a biproduct of this discussion, new examples of infinite dimensional Virasoro-type Lie algebras and their nonlinear analogues are constructed. As an algebro-geometrical application it is shown that WDVV is just the universal system of integrable differential equations (higher order analogue of the Painleve-VI equation) specifying the periods of the Abelian differentials on Riemann surfaces as functions on moduli of these surfaces.