DR/DZ equivalence conjecture and tautological relations

Buryak, Alexandr¹ ; Guéré, Jérémy² ; Rossi, Paolo³

¹ School of Mathematics, University of Leeds, Leeds, United Kingdom
² Université Grenoble Alpes, CNRS, Institut Fourier, Grenoble, France
³ Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova, Padova, Italy

Geometry & topology, Tome 23 (2019) no. 7, pp. 3537-3600.

Résumé

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 ${\overline{\mathcal{ℳ}}}_{g,n+m}$ to ${\overline{\mathcal{ℳ}}}_{g,n}$ and then restricted to ${\mathcal{ℳ}}_{g,n}$ for any $g,n,m\ge 0$. 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 $g\le 2$. 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 $g\le 3$.

Détail

DOI : 10.2140/gt.2019.23.3537

Classification : 14H10, 37K10
Keywords: moduli space of curves, cohomology, double ramification cycle, partial differential equations

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