Geometry & Topology

DR/DZ equivalence conjecture and tautological relations

Alexandr Buryak, Jérémy Guéré, and Paolo Rossi

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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 ̄g,n+m to ̄g,n and then restricted to g,n for any g,n,m0. 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 g2. 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 g3.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H10: Families, moduli (algebraic)
Secondary: 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)

moduli space of curves cohomology double ramification cycle partial differential equations


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.

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