Graph enumeration for moduli spaces of curves and maps

TL;DR

The paper develops a graph-enumeration calculus to compute $S_n$-equivariant motivic invariants of graph-stratified moduli spaces, applying it to $\overline{\mathcal{M}}_{g,n}$ and to torus-fixed stable maps with suitable $\mathbb{C}^\star$-actions. It introduces the Pólya--Petersen character valued in $\Lambda^{[2]}$ and a plethystic action that reduces complex invariants to ordinary symmetric functions and mixed Hodge-structure data, enabling finite graph-sum expressions. The authors derive fixed-genus and all-genus generating functions via sums over graphs decorated by 2-partitions, Adams operations, and grafting (cores and caterpillars), and extend the framework to combinatorial subspaces like compact-type curves and torus-fixed maps through localization-encoded graph sums. The approach connects and extends Getzler--Kapranov’s modular operad viewpoint, providing a versatile, computable toolkit for Euler characteristics and Hodge-structure data across moduli problems by leveraging graph combinatorics and representation theory of wreath products.

Abstract

We develop a calculus based on graph enumeration for $S_n$-equivariant motivic invariants of graphically stratified moduli spaces. We apply our theory to the Deligne--Mumford moduli space $\overline{\mathcal{M}}_{g, n}$ and to the space of torus-fixed stable maps $\overline{\mathcal{M}}_{g, n}(X, β)^{\mathbb{C}^\star}$ when the target $X$ admits an appropriate $\mathbb{C}^\star$-action, deriving new formulas in each case. A key role is played by the Pólya--Petersen character of a graph, which enriches Pólya's classical cycle index polynomial. This character is valued in an algebra $Λ^{[2]}$ of wreath product symmetric functions, which we study from combinatorial and representation-theoretic perspectives. We prove that this algebra may be viewed as the Grothendieck ring of the category of polynomial functors which take symmetric sequences of vector spaces to vector spaces, building on foundational work of Macdonald. This leads to a plethystic action of $Λ^{[2]}$ on the ring $Λ$ of ordinary symmetric functions. Using this action, we derive our formulas, which ultimately involve only ordinary symmetric functions and the Grothendieck ring of mixed Hodge structures.

Figures (4)

• Figure 1: Computing $\overline{\mathsf{a}}_2$ as a sum over graphs. The formula in Theorem \ref{['thm:fixed-genus-thm']} is obtained by summing each graph contribution, and then taking plethysm with $p_1 + \overline{\mathsf{a}}_0'$. In the table we have set $\mathsf{c_1}:= \frac{1}{1 - \mathsf{a}_0"}$ and $\mathsf{c}_2:= \frac{1 + 2 \dot{\mathsf{a}}_0}{1 - \psi_2(\mathsf{a}_0")}$.
• Figure 2:
• Figure 3: The generating function $\mathsf{a}^{(0)}_3$ is computed by taking the sum of each of the graph contributions in the table, and then performing plethysm with $p_1 + \overline{\mathsf{a}}_0'$. As in Figure \ref{['fig:genus-two-fig']}, we set $\mathsf{c}_1 = \frac{1}{1 - \mathsf{a}_0"}$ and $\mathsf{c}_2 = \frac{1 + 2 \dot{\mathsf{a}}_0}{1 - \psi_2(\mathsf{a_0")}}$.
• Figure 4: The generating function $\overline{\mathsf{a}}_{\mathbb{P}^r, 3}^{\mathbb{C}^\star}$ is obtained by summing each of the graph contributions in the table.

Theorems & Definitions (183)

• Remark 1.2
• Example 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Definition 1.8
• Theorem
• Definition 1.9
• Theorem 1.10