DR/DZ equivalence conjecture and tautological relations

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 $g,n,m\ge 0$. 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 $g\le 3$.

v2: 64 pages, typos corrected, accepted in Geometry and Topology, adapted to the journal style

References in corpus (5)

• Double ramification cycles on the moduli spaces of curves
• Double ramification cycles and integrable hierarchies
• Tau-structure for the Double Ramification Hierarchies
• Integrable systems of double ramification type
• Integrability, Quantization and Moduli Spaces of Curves

Cited by in corpus (8)

• Double ramification cycles with target varieties
• Flat F-manifolds, F-CohFTs, and integrable hierarchies
• Quadratic double ramification integrals and the noncommutative KdV hierarchy
• Towards a bihamiltonian structure for the double ramification hierarchy
• The quantum Witten-Kontsevich series and one-part double Hurwitz numbers
• The master relation for polynomiality and equivalences of integrable systems
• Quantum KdV hierarchy and shifted symmetric functions
• Stable tree expressions with Omega-classes and Double Ramification cycles