Tautological relations and integrable systems (Q6633225)

Let \(\overline{\mathcal{M}}_{g,n}\) be the moduli spaces of stable algebraic curves of genus \(g\) with \(n\) marked points. \NIn this paper, for any \(m\geq0\), the authors present a family of conjectural relations in the cohomology of \(\overline{\mathcal{M}}_{g,n+m}\) parameterized by integers \(d_1,\dots,d_n\geq 0\) satisfying \(\sum d_i\geq 2g+m-1\). For fixed \(g,n,m, d_1,\dots,d_n\), the conjectural relation lies in \(H^{2\sum d_i}(\overline{\mathcal{M}}_{g,n+m},\mathbb{C})\). They explain that these conjectural relations naturally imply some fundamental properties of the double ramification (DR) and the Dubrovin-Zhang (DZ) hierarchies. A large part of these relations has a surprisingly simple form: the tautological classes involved in the relations are given by stable graphs that are trees and that are decorated only by powers of the psi-classes at half-edges. \NThe authors show that the proposed conjectural relations imply certain fundamental properties of the DZ and the DR hierarchies associated to F-cohomological field theories. The relations naturally extend a similar system of conjectural relations, which were proposed in an earlier work of the first author et al. [Geom. Topol. 23, No. 7, 3537--3600 (2019; Zbl 1469.14065)] and which are responsible for the normal Miura equivalence of the DZ and the DR hierarchy associated to an arbitrary cohomological field theory. \NFinally, the authors prove all the above mentioned relations in the case \(n=1\) and arbitrary \(g\) using a variation of the method from a paper by \textit{X. Liu} and \textit{R. Pandharipande} [J. Algebr. Geom. 20, No. 3, 479--494 (2011; Zbl 1223.14066)], this can be of independent interest. In particular, this proves the main conjecture from their previous joined work [the authors et al., Épijournal de Géom. Algébr., EPIGA 6, Article 8, 17 p. (2022; Zbl 1487.14065)]. They also prove all the above mentioned relations in the case \(g=0\) and arbitrary \(n\). \N\NThis work consists of the following basic parts: 1) Introduction. 2) Conjectural cohomological relations. 3) An equivalent formulation of the conjectures. 4) Conjectural relations and the fundamental properties of the DR and the DZ hierarchies. 5) Proof of Theorem 2.2 (the three conjectures are true for \(n=1\)). 6) Proof of Theorem 2.3 (the three conjectures are true for \(g=0\)). 7) A reduction of the system of relations in the case \(m=2\). Appendix. Localization in the moduli space of stable relative maps.