DR Hierarchies: From the Moduli Spaces of Curves to Integrable Systems

The main goal of the paper is to show that the DR hierarchies, introduced by the author in an earlier paper, allow one to establish, in the most clear way, a relation between the topology of the Deligne–Mumford compactification $\overline {\mathcal M}_{g,n}$ of the moduli space $\mathcal M_{g,n}$ of smooth algebraic curves of genus $g$ with $n$ marked points and integrable systems of mathematical physics. We will also discuss a promising approach given by the theory of DR hierarchies to the solution of a general problem in the area of Witten-type conjectures, namely, to the proof of the existence of a Dubrovin–Zhang hierarchy for an arbitrary cohomological field theory.