# DR/DZ equivalence conjecture and tautological relations

## TL;DR

This paper introduces conjectural tautological relations in the moduli space of stable curves that imply the DR/DZ equivalence conjecture, providing new formulas for certain classes and verifying them in low genus cases.

## Contribution

It proposes a family of tautological relations implying the DR/DZ equivalence and proves their validity in low genus, also deriving a new formula for the class λ_g.

## Key findings

- Relations hold in genus 0 and 1
- Relations verified after certain pushforward and restriction
- New formula for λ_g class checked for g ≤ 3

## Abstract

In this paper 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. 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 $\overline{\mathcal{M}}_{g,n+m}$ to $\overline{\mathcal{M}}_{g,n}$ and then restricted to $\mathcal{M}_{g,n}$, for any $g,n,m\geq 0$. 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 $g\leq 2$. As an application we find a new formula for the class $\lambda_g$ as a linear combination of dual trees intersected with kappa and psi classes, and we check it for $g \leq 3$.