Graph enumeration for moduli spaces of curves and maps

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 $\overline{\mathcal{M}}_{g, n}(X, \beta)^{\mathbb{C}^\star}$ when the target $X$ admits an appropriate $\mathbb{C}^\star$-action, deriving new formulas in each case. A key role is played by the P\'olya--Petersen character of a graph, which enriches P\'olya's classical cycle index polynomial. This character is valued in an algebra $\Lambda^{[2]}$ 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 $\Lambda^{[2]}$ on the ring $\Lambda$ 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.