Tautological relations and integrable systems

TL;DR

The paper proposes a broad family of conjectural tautological relations in the cohomology of the moduli spaces $\overline{\mathcal{M}}_{g,n}$, organized around tree graphs decorated only by powers of $\psi$-classes. It shows these relations govern deep structural properties of the DR and DZ integrable hierarchies attached to $F$-CohFTs, extending prior Miura- and normal Miura-type relations. The authors prove the conjectures in the cases $n=1$ for arbitrary $g$ and in genus zero for arbitrary $n$, and develop equivalent formulations that reduce the problem to a finite set of degree-$2g+m-1$ relations. They also establish a reduction principle for $m\ge 2$, connect these relations to localization methods on moduli spaces of stable relative maps, and provide a geometric framework for polynomiality of the DZ hierarchy and DR/DZ equivalences in the broader setting of partial CohFTs. The work unifies geometric tautological relations with integrable-system structures and offers concrete tools for verifying DR/DZ equivalences across broad families of CohFTs.

Abstract

We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus $g$ with $n$ marked points. 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. We show that the proposed conjectural relations imply certain fundamental properties of the Dubrovin-Zhang (DZ) and the double ramification (DR) hierarchies associated to F-cohomological field theories. Our relations naturally extend a similar system of conjectural relations, which were proposed in an earlier work of the first author together with Guéré and Rossi and which are responsible for the normal Miura equivalence of the DZ and the DR hierarchy associated to an arbitrary cohomological field theory. Finally, we prove all the above mentioned relations in the case $n=1$ and arbitrary $g$ using a variation of the method from a paper by Liu and Pandharipande, this can be of independent interest. In particular, this proves the main conjecture from our previous joined work together with Hernández Iglesias. We also prove all the above mentioned relations in the case $g=0$ and arbitrary $n$.

Theorems & Definitions (55)

• Example 1.1
• Definition 2.1
• Conjecture 1
• Conjecture 2
• Conjecture 3: cf. BGR19
• Theorem 2.2
• Theorem 2.3
• Definition 3.1
• Example 3.2
• Remark 3.3