A new proof of the \(\lambda_g\) conjecture in genus 2 (Q6596270)

In modern enumerative geometry, \(\lambda_g\) typically denotes the top Chern class of the Hodge bundle \(\mathbb{E}\) on the compactification of the moduli of stable curves \(\overline{M}_{g,n}\). Efforts to understand intersection numbers on these moduli spaces connect deeply with Gromov-Witten theory, Hurwitz theory, and topological string theory. The \(\lambda_g\) conjecture in particular considers the integral of \(\lambda_g\) together with the \(\psi\)-classes on these moduli spaces and provides an explicit formula. It was first found as a consequence of the Virasoro conjecture by \textit{E. Getzler} and \textit{R. Pandharipande} [Nucl. Phys., B 530, No. 3, 701--714 (1998; Zbl 0957.14038)] and was later established by \textit{C. Faber} and \textit{R. Pandharipande} [Ann. Math. (2) 157, No. 1, 97--124 (2003; Zbl 1058.14046)] by virtual localization.\N\NIn more recent progress, \textit{F. Janda} et al. [Publ. Math., Inst. Hautes Étud. Sci. 125, 221--266 (2017; Zbl 1370.14029)] studied the double ramification cycles on \(\overline{M}_{g,n}\). These cycles parametrize genus \(g\) curves \(C\) that admit a finite map \(C \to \mathbb{P}^1\) ramified over \(0\) and \(\infty\), with the ramification data encoded in a vector. In particular, they proved Pixton's formula, showing that the locus of the double ramification cycle can be expressed explicitly as a polynomial. As a special case, the Hodge classes \(\lambda_g\) arise as double ramification loci (up to a sign), and the paper of \textit{S. Molcho} et al. [Compos. Math. 159, No. 2, 306--354 (2023; Zbl 1523.14055)] explicitly discusses in Section 6.4 the generation of the Hodge class \(\lambda_g\) up to \(g=4\).\N\NThis paper under review studied the case of \(\lambda_2\) with Pixton's formula, showing that the Hodge class \(\lambda_2\) can be expressed as a linear combination of two specific boundary strata with a particular dual graph configuration (c.f. formula (2)).\N\NThis paper under review confirmed the \(\lambda_g\) conjecture for genus \(2\) using these two strata. More specifically, the two strata can be expressed either by a gluing morphism and the projection formula or by an induction on the number of marked points. In particular, the authors deduce the inductive process in their Lemma 3.1, where the boundary strata integrals relate the cases of \(n+1\) and \(n\) marked points. Together with the dilaton and string equations these two calculations yield the final result in Theorem 3.2.\N\NIn summary, the authors use a particular expression of the \(\lambda_g\) class by the double ramification locus to calculate its integral in terms of a special configuration of boundary strata. Their study of the combinatorial properties of these boundary strata proves the \(\lambda_g\) conjecture, providing a concise, elementary way to understand this conjecture in lower genera. Similar arguments in genus \(3\) and \(4\) may be carried out with further analysis.