A conjectural formula for $DR_g(a,-a) λ_g$

TL;DR

The paper proposes a concise conjectural formula for $DR_g(a,-a)\lambda_g$ in the tautological ring $R^{2g}(\overline{\mathcal{M}}_{g,2})$, refining the one-point case of a prior conjecture and showing independence from $a$ by establishing the key identity $a^{-2g}\mathrm{DR}_g(a,-a)\lambda_g={\mathsf{B}}^g$. It defines ${\mathsf{B}}^g$ via bamboo-type strata built from psi-classes and a gluing operation, then proves a suite of properties using Liu–Pandharipande relations, including symmetry, boundary-intersection vanishing, and psi-class pull-backs. A central contribution is proving the equivalence between the new conjectural formula and the earlier push-forward conjecture from BGR, by deriving a common recursive framework that determines the class from either side. The results provide a compact, verifiable expression for the DR-lambda class in genus $g$ and illuminate structural relations in the tautological ring, potentially guiding future proofs and computations in the study of moduli of curves. The work links two conjectural pictures and highlights the role of boundary strata and psi-classes in understanding the double ramification cycle with Hodge bundle contributions.

Abstract

We propose a conjectural formula for $DR_g(a,-a) λ_g$ and check all its expected properties. Our formula refines the one point case of a similar conjecture made by the first named author in collaboration with Guéré and Rossi, and we prove that the two conjectures are in fact equivalent, though in a quite non-trivial way.

Theorems & Definitions (24)

• Conjecture 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• Remark 2.6
• Proposition 3.1: Liu-Pand
• Corollary 3.2
• Corollary 3.3: Liu-Pand