DR/DZ equivalence conjecture and tautological relations

Buryak, Alexandr; Guéré, Jérémy; Rossi, Paolo

doi:10.2140/gt.2019.23.3537

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DR/DZ equivalence conjecture and tautological relations

Alexandr Buryak, Jérémy Guéré and Paolo Rossi

Geometry & Topology 23 (2019) 3537–3600

DOI: 10.2140/gt.2019.23.3537

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 ${\bar{ℳ}}_{g, n + m}$ to ${\bar{ℳ}}_{g, n}$ and then restricted to $ℳ_{g, n}$ 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 $λ_{g}$ as a linear combination of dual trees intersected with kappa- and psi-classes, and we check it for $g \leq 3$ .

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Keywords

moduli space of curves, cohomology, double ramification cycle, partial differential equations

Mathematical Subject Classification 2010

Primary: 14H10

Secondary: 37K10

References

Publication

Received: 5 May 2018

Accepted: 4 November 2018

Published: 30 December 2019

Proposed: Yasha Eliashberg

Seconded: Richard Thomas, Jim Bryan

Authors

Alexandr Buryak
School of Mathematics University of Leeds Leeds United Kingdom
https://sites.google.com/site/alexandrburyakhomepage/home
Jérémy Guéré
Université Grenoble Alpes CNRS Institut Fourier Grenoble France
https://www-fourier.ujf-grenoble.fr/~guerej/index.html
Paolo Rossi
Dipartimento di Matematica “Tullio Levi-Civita” Università degli Studi di Padova Padova Italy
http://www.math.unipd.it/~rossip/

Volume 23, issue 7 (2019)