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

1. Problem Statement

The paper addresses the computation of Faber's socle intersection numbers within the tautological ring of the moduli space of curves, $\mathcal{M}_{g,n}$. Specifically, it seeks to evaluate integrals of the form:
$$\int_{\mathcal{M}_{g,n}} \lambda_g \lambda_{g-1} \prod_{i=1}^n \psi_i^{d_i}$$
where $d_1 + \dots + d_n = g - 2 + n$, $\lambda_i$ are Chern classes of the Hodge bundle, and $\psi_i$ are cotangent line classes.

While Faber's formula for these numbers was originally conjectured in 1999 and has since been proven via five distinct methods (Virasoro constraints, Mumford's formula, double ramification cycles, 3-spin relations, and half-spin relations), the authors aim to provide a sixth proof. This new proof is significant not merely for verification, but because it relies on a completely different set of geometric ideas derived from the Gromov–Witten (GW) theory of the elliptic curve and yields a new tautological relation useful for analyzing quantum integrable systems.

2. Methodology

The authors employ a synthesis of modern GW theory, quasimodular forms, and the geometry of double ramification (DR) cycles. The methodology proceeds in three main stages:

A. Derivation of a New Tautological Relation

The core of the proof relies on a recent result by Oberdieck and Pixton regarding the GW theory of the elliptic curve.

1. Cohomological Field Theory (CohFT): The authors consider the classes $c_{g,m}(p^{\otimes m})$ associated with the GW theory of the elliptic curve, where all primary fields correspond to the point class.
2. Holomorphic Anomaly Equation: Utilizing the holomorphic anomaly equation, Oberdieck and Pixton established an explicit formula for $c_{g,m}(p^{\otimes m})\lambda_{g-1}$ involving quasimodular forms (specifically Eisenstein series $G_{2g}$ and the operator $q\partial_q$).
3. Graph Expansion: The authors compare the analytic formula with the graph expansion of the CohFT classes provided by Oberdieck and Pixton. By analyzing the contributions of stable graphs, they isolate a specific subset of graphs: necklace graphs (cycles of vertices with one marked point each).
4. Simplification: Through a refinement of the argument in [OP23], they demonstrate that for the specific weight grading required, only these necklace graphs contribute non-trivially. This leads to the derivation of Proposition 2.1, a new tautological relation in $H^*(\mathcal{M}_{g,m})$ expressing a sum over necklace graphs (decorated with DR cycles) as a product of $\lambda$-classes, $\psi$-classes, and a sum of reciprocals of $\psi$-classes.

B. Intersection with Double Ramification Cycles

To extract numerical values from the new tautological relation, the authors must compute intersection numbers involving DR cycles.

1. Recursion Relations: They utilize a known recursion relation for $\psi$-classes on DR cycles (derived from [BSSZ15] and related to the KdV hierarchy Hamiltonians).
2. Explicit Computation: In Proposition 3.1, they prove a closed-form formula for the integral:
   $$\int_{\mathcal{M}_{g,3}} DR_g(-a_1-a_2, a_1, a_2) \lambda_g \psi_1^g$$
   This formula is a polynomial in $a_1, a_2$ involving Bernoulli numbers and double factorials.

C. Synthesis

The final step involves substituting the explicit intersection numbers from Proposition 3.1 into the new tautological relation (Proposition 2.1). By carefully matching the coefficients of the quasimodular forms and the combinatorial factors arising from the necklace graphs, the authors recover Faber's original formula.

3. Key Contributions

1. New Tautological Relation (Proposition 2.1):
   The paper establishes a novel identity in the tautological ring:
   $$\frac{1}{(m-1)!} \sum_{\Gamma \in \vec{\mathcal{N}}} (b_{\Gamma, g})_* \left( \bigotimes_{i=1}^m DR_{g(v_i)}(0, 1, -1) \lambda_{g(v_i)} \right) = \text{RHS involving } \lambda_g, \lambda_{g-1}, \psi_i$$
   This relation connects the geometry of necklace graphs decorated with DR cycles to the fundamental classes of the moduli space.

2. Alternative Proof of Faber's Formula:
   The paper provides a rigorous derivation of Faber's formula:
   $$\int_{\mathcal{M}_{g,n}} \lambda_g \lambda_{g-1} \prod_{i=1}^n \psi_i^{d_i} = (-1)^{g-1} \frac{B_{2g} (2g-3+n)!}{2^{2g-1} (2g)!} \prod_{i=1}^n \frac{1}{(2d_i-1)!!}$$
   This proof is distinct from previous ones as it stems from the GW theory of the elliptic curve rather than Virasoro constraints or spin curves.

3. Explicit DR Intersection Formula:
   The authors provide a new explicit formula (Proposition 3.1) for the intersection of DR cycles with $\lambda_g \psi_g$ on $\mathcal{M}_{g,3}$, which serves as a crucial computational tool for the main proof.

4. Connection to Integrable Systems:
   The authors note that the new tautological relation and the underlying ideas are instrumental in the analysis of the quantum integrable systems of Buryak–Rossi, specifically relating to the Hamiltonians of the KdV hierarchy.

4. Results

• Verification: The paper successfully verifies Faber's formula using the machinery of the elliptic curve GW theory.
• Combinatorial Simplification: The proof demonstrates that in the context of the specific weight grading $2g-2+2m$, the complex graph sum from the GW theory collapses entirely to a sum over oriented necklace graphs.
• Quasimodular Form Connection: The work explicitly links the topological recursion of the moduli space to the operator $(q\partial_q)^{m-1} G_{2g}(q)$, showing how the derivative of Eisenstein series encodes the intersection numbers.

5. Significance

• Unification of Fields: The paper bridges the gap between the GW theory of the elliptic curve (a topic in mathematical physics and string theory) and the classical intersection theory of moduli spaces of curves.
• New Tools for Tautological Rings: The derived tautological relation (Proposition 2.1) is a "beautiful" new relation that was previously unknown. It offers a new structural perspective on the tautological ring, potentially simplifying future computations.
• Application to Quantum Integrable Systems: The authors emphasize that this proof is not just an alternative derivation but a necessary step for understanding the quantum integrable systems associated with the double ramification hierarchy. The "new set of powerful ideas" introduced here is expected to be more versatile for these specific applications than previous methods.
• Refinement of Oberdieck–Pixton's Work: The paper acts as a refinement of the Oberdieck–Pixton results, extracting specific geometric consequences (the necklace relation) that were implicit in their broader GW theory framework.

In summary, this short note provides a sophisticated, physics-inspired proof of a classical algebraic geometry result, introducing a new tautological relation that deepens the understanding of the structure of $\mathcal{M}_{g,n}$ and its connections to integrable systems.