The master relation for polynomiality and equivalences of integrable systems
Xavier Blot, Adrien Sauvaget, Sergey Shadrin
TL;DR
This work proves the master relation in the tautological ring $R^*({\overline{\mathcal{M}}}_{g,n+m})$, for $g\ge0$, $n,m\ge1$, with $2g-2+n+m>0$, as conjectured in prior work. The master relation, established in the Gorenstein quotient, implies polynomiality properties for the Dubrovin-Zhang hierarchies of various CohFT variants and their equivalence to double ramification hierarchies (BDGR1/BDGR20). The proof uses virtual localization of relative stable maps to $\mathbb{P}^1$, reducing the calculation to contributions from pre-stable star rooted trees and employing Mumford's relations to align terms. Consequently, the paper resolves conjectures linking tautological relations to integrable-system structures arising from topological field theories and clarifies how polynomiality and DR/DZ equivalences emerge from the master relation.
Abstract
We prove the so-called master relation in the tautological ring of the moduli space of curves that implies polynomial properties of the Dubrovin-Zhang hierarchies associated to different versions of cohomological field theories as well as their equivalences to the corresponding double ramification hierarchies.