DR/DZ equivalence conjecture and tautological relations

We present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin–Zhang equivalence conjecture introduced by the authors with Dubrovin (Comm. Math. Phys. 363 (2018) 191–260). Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ramification cycle. We prove that both families behave the same way upon pullback and pushforward with respect to forgetting a marked point. We also prove that our conjectural relations are true in genus 0 and 1 and also when first pushed forward from ˉMg,n+m to ˉMg,n and then restricted to Mg,n for any g,n,m≥0. Finally we show that, for semisimple CohFTs, the DR/DZ equivalence only depends on a subset of our relations, finite in each genus, which we prove for g≤2. As an application we find a new formula for the class λg as a linear combination of dual trees intersected with kappa- and psi-classes, and we check it for g≤3.