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Faber's socle intersection numbers via Gromov--Witten theory of elliptic curve

Xavier Blot, Sergey Shadrin, Ishan Jaztar Singh

TL;DR

The paper addresses Faber's socle intersection numbers in the tautological ring of $\overline{\mathcal{M}}_{g,n}$ and presents a new proof based on a tautological relation derived from the Gromov–Witten theory of the elliptic curve (Oberdieck–Pixton) combined with explicit double ramification cycle calculations. It develops a necklace-graph calculus to express a boundary relation that equates a sum over oriented necklace graphs of DR-cycle–weighted boundary pushforwards with a universal class involving $\lambda_g$, $\lambda_{g-1}$, and $\psi$-classes, with the computation anchored by quasimodular forms in $\mathrm{QMod}$. Intersection-theoretic recursion for DR cycles with $\lambda_g$ and $\psi$-classes is established, enabling a weight-filtered top-degree analysis tied to KdV Hamiltonian recursions. The resulting corollary reproduces Faber's socle formula, linking tautological relations, DR calculus, and elliptic-GW quasimodular structures, and suggesting new tautological relations relevant to quantum integrable systems.

Abstract

The goal of this very short note is to give a new proof of Faber's formula for the socle intersection numbers in the tautological ring of $\mathcal{M}_g$. This new proof exhibits a new beautiful tautological relation that stems from the recent work of Oberdieck--Pixton on the Gromov--Witten theory of the elliptic curve via a refinement of their argument, and some straightforward computation with the double ramification cycles that enters the recursion relations for the Hamiltonians of the KdV hierarchy.

Faber's socle intersection numbers via Gromov--Witten theory of elliptic curve

TL;DR

The paper addresses Faber's socle intersection numbers in the tautological ring of and presents a new proof based on a tautological relation derived from the Gromov–Witten theory of the elliptic curve (Oberdieck–Pixton) combined with explicit double ramification cycle calculations. It develops a necklace-graph calculus to express a boundary relation that equates a sum over oriented necklace graphs of DR-cycle–weighted boundary pushforwards with a universal class involving , , and -classes, with the computation anchored by quasimodular forms in . Intersection-theoretic recursion for DR cycles with and -classes is established, enabling a weight-filtered top-degree analysis tied to KdV Hamiltonian recursions. The resulting corollary reproduces Faber's socle formula, linking tautological relations, DR calculus, and elliptic-GW quasimodular structures, and suggesting new tautological relations relevant to quantum integrable systems.

Abstract

The goal of this very short note is to give a new proof of Faber's formula for the socle intersection numbers in the tautological ring of . This new proof exhibits a new beautiful tautological relation that stems from the recent work of Oberdieck--Pixton on the Gromov--Witten theory of the elliptic curve via a refinement of their argument, and some straightforward computation with the double ramification cycles that enters the recursion relations for the Hamiltonians of the KdV hierarchy.

Paper Structure

This paper contains 5 sections, 3 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. New tautological relation
  4. Intersection numbers with the double ramification cycles
  5. A new proof of the socle intersection numbers

Key Result

Proposition 2.1

We have the following relation in $H^*(\overline{\mathcal{M}}_{g,m})$: Here $\mathsf{b}_{\Gamma,g}\colon \prod_{i=1}^m \overline{\mathcal{M}}_{g(v_i),3} \to \overline{\mathcal{M}}_{g,m}$ is the boundary map, $\mathrm{DR}_{g(v_i)}(0,1,-1)$ is the double ramification cycle on $\overline{\mathcal{M}}_{g(v_i),3}$ assigned to $v_i$, where the multiplicity $0$ is assigned to

Theorems & Definitions (6)

  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Corollary 4.1
  • proof