## Geometry & Topology

### DR/DZ equivalence conjecture and tautological relations

#### Abstract

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 $ℳ̄g,n+m$ to $ℳ̄g,n$ 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 $g≤3$.

#### Article information

Source
Geom. Topol., Volume 23, Number 7 (2019), 3537-3600.

Dates
Received: 5 May 2018
Accepted: 4 November 2018
First available in Project Euclid: 7 January 2020

Permanent link to this document
https://projecteuclid.org/euclid.gt/1578366034

Digital Object Identifier
doi:10.2140/gt.2019.23.3537

Mathematical Reviews number (MathSciNet)
MR4059088

Zentralblatt MATH identifier
07152164

#### Citation

Buryak, Alexandr; Guéré, Jérémy; Rossi, Paolo. DR/DZ equivalence conjecture and tautological relations. Geom. Topol. 23 (2019), no. 7, 3537--3600. doi:10.2140/gt.2019.23.3537. https://projecteuclid.org/euclid.gt/1578366034

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