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Structures in topological recursion relations

Felix Janda, Xin Wang

TL;DR

This work analyzes degree-$g$ topological recursion relations on the moduli space $\overline{\mathcal{M}}_{g,n}$, revealing the bouquet-class coefficient and linear relations for rational tails, and develops an algorithmic approach to derive universal equations for Gromov–Witten invariants. It integrates the tautological-ring framework, Pixton’s double ramification formula, and stable-graph calculus to produce concrete TRR expressions and a recursive scheme for intersection numbers. Key results include a precise bouquet-coefficient formula tied to loop graphs, a strengthening conjecture on uniqueness, and a recursive method for computing $\int_{\overline{\mathcal{M}}_{g,n}}\prod_i \psi_i^{k_i}$ via Pixton’s framework. Applications span universal GW equations, DR formulas for $\lambda_g$, and a practical recursion for tautological intersection numbers, enhancing both theoretical understanding and computational reach in the tautological setting.

Abstract

In this paper, we study the basic structures of degree-$g$ topological recursion relations on the moduli space of curves $\overline{\mathcal{M}}_{g,n}$: (i) The coefficient of the bouquet class on $\overline{\mathcal{M}}_{g,n}$, which gives the answer to a conjecture of T. Kimura and X. Liu; (ii) Linear relations among the coefficients of certain rational tails locus of $\overline{\mathcal{M}}_{g,n}$. Three applications of topological recursion relations will be discussed: (i) Coefficients of universal equations for Gromov-Witten invariants for any smooth projective variety; (ii) The coefficient of the bouquet class in the double ramification formula of the top Hodge class $λ_g$; (iii) A new recursive formula for computing the intersection numbers on the moduli space of stable curves.

Structures in topological recursion relations

TL;DR

This work analyzes degree- topological recursion relations on the moduli space , revealing the bouquet-class coefficient and linear relations for rational tails, and develops an algorithmic approach to derive universal equations for Gromov–Witten invariants. It integrates the tautological-ring framework, Pixton’s double ramification formula, and stable-graph calculus to produce concrete TRR expressions and a recursive scheme for intersection numbers. Key results include a precise bouquet-coefficient formula tied to loop graphs, a strengthening conjecture on uniqueness, and a recursive method for computing via Pixton’s framework. Applications span universal GW equations, DR formulas for , and a practical recursion for tautological intersection numbers, enhancing both theoretical understanding and computational reach in the tautological setting.

Abstract

In this paper, we study the basic structures of degree- topological recursion relations on the moduli space of curves : (i) The coefficient of the bouquet class on , which gives the answer to a conjecture of T. Kimura and X. Liu; (ii) Linear relations among the coefficients of certain rational tails locus of . Three applications of topological recursion relations will be discussed: (i) Coefficients of universal equations for Gromov-Witten invariants for any smooth projective variety; (ii) The coefficient of the bouquet class in the double ramification formula of the top Hodge class ; (iii) A new recursive formula for computing the intersection numbers on the moduli space of stable curves.
Paper Structure (30 sections, 26 theorems, 177 equations, 1 figure)

This paper contains 30 sections, 26 theorems, 177 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structures of topological recursion relations
  3. Uniqueness of the coefficients in topological recursion relations
  4. Universal equations for Gromov--Witten invariants
  5. Coefficient of the bouquet class in the DR formula of the Hodge class $\lambda_g$
  6. Intersection numbers on moduli space of curves
  7. The tautological ring and topological recursion relations
  8. The strata algebra and the tautological ring of $\overline{\mathcal{M}}_{g,n}$
  9. Pixton's DR formula
  10. Polynomiality properties
  11. An algorithm for deriving topological recursion relations on $\overline{\mathcal{M}}_{g,n}$
  12. Generating series formula for $\Omega_{g, M}$
  13. The push-forward formula of $\pi_*(\Omega_{g,M})$
  14. The coefficient of the bouquet class in topological recursion relations
  15. Contributions of stable graphs to $\pi_* \Omega_{g, M}$
  16. ...and 15 more sections

Key Result

Theorem 1

For any non-negative integers $\{k_i\}_{i=1}^{n}$ satisfying $\sum_{i=1}^{n}k_i=g\geq1$, there exists a topological recursion relation on $\overline{\mathcal{M}}_{g,n}$ of the form The omitted terms in the above formula satisfy the following condition: there are no $\kappa$ classes and no genus-$h$ vertices with monomial of $\psi$ classes of degree $\geq h+\delta_{h}^0$ for $0\leq h\leq g$.

Figures (1)

  • Figure 1: Dual graphs for $\overline{\mathcal{M}}_{1,2}$

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Conjecture 3
  • Corollary 4: Corollary \ref{['cor:bouquet-correlation-fun']}
  • Corollary 5: Corollary \ref{['cor:two-pt-correlation-fun']}
  • Proposition 6: Proposition \ref{['prop:lambda_g']}
  • Theorem 7: Theorem \ref{['thm:algo-inter']}
  • Theorem 8: clader2018pixton
  • Theorem 9: Pixton Pixton2024, Spelier spelier2024polynomiality
  • Corollary 10
  • ...and 33 more