DR hierarchies: from the moduli spaces of curves to integrable systems (Q6633146)

The central object of this paper is the moduli space \(\mathcal{M}_{g,n}\) of smooth algebraic curves of genus \(g\) with \(n\) marked points, as well as its compactification \(\overline{\mathcal{M}}_{g,n}\), constructed by \textit{P. Deligne} and \textit{D. Mumford} [Publ. Math., Inst. Hautes Étud. Sci. 36, 75--109 (1969; Zbl 0181.48803)] using stable curves, which are marked projective curves that have at most nodal singularities, whose marked points lie on the smooth part of the curve, and whose automorphism group is finite. \N\NThe author asks the questions whether is it true that the correlators of an arbitrary cohomological field theory (CohFT) are described by some integrable system and what geometric properties of the moduli spaces \(\overline{\mathcal{M}}_{g,n}\) are responsible for the appearance of integrable systems of PDEs. In this paper, the author wants to convince the reader that the theory of the double ramification (DR) hierarchies, proposed by the author in his paper [Commun. Math. Phys. 336, No. 3, 1085--1107 (2015; Zbl 1329.14103)], gives a satisfactory answer to this problem. Moreover, the theory of DR hierarchies gives a promising approach to a proof of the existence of a Dubrovin-Zhang hierarchy for an arbitrary CohFT. \N\NThe aim of this paper is to show that the DR hierarchies, introduced by the author in the above-mentioned 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. The author discusses 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. This work consists of the following basic parts: 1) Introduction. 2) Module spaces of stable curves. 3) Cohomological field theories. 4) Formal theory of evolutionary partial differential equations. 5) DR Hierarchies. 6) Dubrovin-Zhang hierarchies and equivalence conjecture.