Structures in topological recursion relations

Technical Summary: Structures in Topological Recursion Relations

Problem Statement
The paper investigates the structural properties of degree-$g$ topological recursion relations (TRRs) on the moduli space of stable curves $\mathcal{M}_{g,n}$. While it is established that TRRs exist to express any degree-$g$ polynomial in $\psi$-classes in terms of boundary classes (classes supported on the boundary $\overline{\mathcal{M}}_{g,n} \setminus \mathcal{M}_{g,n}$), explicit formulas for these relations are known only for low genera ($g \leq 4$) and are generally highly complex. The authors aim to uncover fundamental structural constraints on the coefficients of these relations, specifically focusing on two types of boundary strata: the "bouquet class" (a single genus-0 vertex with $g$ loops) and "rational tail" classes (strata where a genus-$g$ curve is attached to a rational component). Understanding these coefficients is crucial because TRRs translate into universal equations for Gromov–Witten invariants, which are essential for computing higher-genus invariants and studying integrable systems like the Virasoro conjecture.

Methodology
The authors employ a constructive algorithm derived from Pixton's double ramification (DR) cycle formula, as refined in their previous work with Clader and Zakharov [5]. The core methodology involves:

1. Pixton's DR Formula: Utilizing the polynomiality of Pixton's class $D_{g,n}(a_1, \dots, a_n)$ in the variables $a_i$ and the parameter $r$. Specifically, they extract coefficients of specific monomials from the degree-$g$ part of these classes to generate TRRs.
2. Strata Algebra and Push-forwards: Working within the strata algebra $S_{g,n}$, they analyze the push-forwards of these classes under forgetful maps. They decompose the contributions of stable graphs into three types based on how they map to the base graph: adding a leg, splitting a leg into a rational tail, or splitting an edge.
3. Generating Functions and Combinatorial Identities: To compute specific coefficients, the authors introduce a linear functional $F$ acting on polynomials of $\psi$-classes. By applying this functional to the contributions of specific graphs (particularly the trivial graph and the bouquet graph), they derive identities involving hyperbolic functions ($\sinh, \cosh$) and Bernoulli numbers.
4. Induction and Linear Relations: They use induction on the genus and the number of marked points, combined with linear relations among coefficients derived from Faber's intersection number conjecture, to isolate specific terms in the TRRs.

Key Contributions and Results

1. The Coefficient of the Bouquet Class (Theorem 1):
   The paper proves a conjecture by T. Kimura and X. Liu regarding the coefficient of the bouquet class in degree-$g$ TRRs. For any non-negative integers $\{k_i\}$ summing to $g$, the coefficient of the class $\xi_{\Gamma_{0,g,[n]}^{loop}}^*(1)$ in the expansion of $\prod \psi_i^{k_i}$ is explicitly given by:
   $$\frac{1}{8^g \prod_{i=1}^n (2k_i + 1)!!}$$
   The authors establish that this coefficient is unique under specific conditions regarding the absence of $\kappa$-classes and high-degree $\psi$-monomials on lower-genus vertices.

2. Linear Relations for Rational Tail Coefficients (Theorem 2):
   The authors derive linear relations among the coefficients of rational tail strata in TRRs. Specifically, for a TRR of the form $\prod \psi_i^{k_i} = \sum a_{\Gamma} \xi_{\Gamma}^*(\dots)$, the sum of coefficients corresponding to rational tails satisfies a specific identity depending on $g$ and $n$. For $n=2$, the coefficient is explicitly calculated as $\frac{(2g-1)!!}{(2k_1-1)!!(2k_2-1)!!}$. For $n \geq 3$, while individual coefficients are not unique, their sum is constrained.

3. Applications to Gromov–Witten Theory (Corollaries 4 & 5):
   Translating the geometric TRRs into the language of Gromov–Witten invariants via the operator $T$ (which shifts descendant levels), the authors provide explicit universal equations.

   • They derive a formula for $\langle\langle T^{k_1}(W_1) \dots T^{k_n}(W_n) \rangle\rangle_g$ involving a sum of genus-0 invariants with specific insertions.
   • They establish a relation for $n=2$ where the genus-$g$ invariant is expressed in terms of a genus-$g$ invariant of a quantum product, modulo lower-genus terms.

4. Coefficient in the Double Ramification Formula for $\lambda_g$ (Proposition 6):
   As an application of Theorem 1, the authors compute the coefficient of the bouquet class in the formula for the top Hodge class $\lambda_g$ (expressed via Pixton's DR formula). They show this coefficient is the coefficient of $t^{2g}$ in the expansion of $\frac{(-1)^g}{2^g} \frac{t^2 e^t}{(e^t-1)^2}$.

5. Recursive Algorithm for Intersection Numbers (Theorem 27):
   The paper proposes a new combinatorial recursive formula for computing intersection numbers $\int_{\mathcal{M}_{g,n}} \prod \psi_i^{k_i}$ where $\sum k_i = 3g-3+n$. Unlike Kontsevich's topological model, this approach uses the degree-$(3g-3+n)$ TRRs derived from Pixton's formula. The algorithm expresses these integrals as a sum over stable graphs with at least one edge, weighted by rational coefficients derived from the DR formula, string equation, and dilaton equation.

Significance and Claims
The paper claims to provide the first explicit structural understanding of the coefficients of degree-$g$ TRRs in arbitrary genus.

• Resolution of Conjectures: It provides an affirmative answer to the Kimura-Liu conjecture on the bouquet class coefficient, a problem that had remained open for general $n$.
• Computational Utility: The derived formulas offer a constructive method to compute specific coefficients that were previously inaccessible due to the complexity of known relations (e.g., Getzler's equation).
• Alternative to Kontsevich: The recursive algorithm for intersection numbers offers a purely combinatorial approach distinct from Kontsevich's matrix integral and topological cell decomposition, potentially providing a new pathway to recover Witten's conjecture.
• Universality: The results yield explicit universal equations for Gromov–Witten invariants of any smooth projective variety, contributing to the broader effort to determine higher-genus invariants from genus-0 data in the semisimple case.

The authors remain modest regarding the uniqueness of the full set of coefficients for rational tails when $n \geq 3$, noting that while linear relations exist, a common factor structure (motivated by Faber's conjecture) for individual coefficients remains an open problem requiring explicit formulae not yet derived.