Schur-Weyl Duality and Higher Abel-Jacobi Invariants for Tautological Cycles in $\mathcal{M}_{g,n}$

TL;DR

This paper explores the Hodge theory of moduli spaces of curves, introducing higher Faber-Pandharipande cycles, and establishing their non-triviality and position within the Leray filtration, linking Schur-Weyl duality to higher Abel-Jacobi invariants.

Contribution

It introduces higher Faber-Pandharipande cycles and connects Schur-Weyl duality with higher Abel-Jacobi invariants in the study of tautological cycles.

Findings

Higher Faber-Pandharipande cycles are non-torsion for large genus.

These cycles lie in depth n+1 of the Leray filtration.

Explicit non-vanishing in specific cohomology components.

Abstract

This article investigates the Hodge theory of the moduli space of genus $g$ curves with $n$ marked points, establishing new connections between Schur-Weyl duality for $\mathfrak{sp}_{g}$ and higher Abel-Jacobi invariants. We develop a represe\\ ntation-theoretic framework that decomposes higher Abel-Jacobi invariants of tautological cycles in $C_{g}^{n}$ according to symplectic Lie algebra representations, leveraging the Leray filtration and motivic decompositions compatible with $\mathfrak{sp}_{2g}$-actions. Central to this work is the introduction of \textbf{higher Faber-Pandharipande cycles} $FP_n = \pi_1^{\times 2}(\Delta_{12}^n \cdot \psi_1)$ in $CH^{n+1}(C_g^2)$, a new family of tautological cycles generalizing classical constructions. We prove these cycles are non-torsion under optimal genus constraints: for families over $(n-1)$-dimensional bases, $FP_n$ is not rationally equivalent to zero when $g \geq 3n+1$. Furthermore, we determine the precise position of $FP_n$ in the Leray filtration of $C_g^2 \to M_g$, showing it lies in depth $n+1$ but no deeper, with explicit non-vanishing in $H^{n+1}(M_g, R^{n+1}f_*\mathbb{Q})$ on the $V_{(n+1,1)}$-isotypic component. This yields the first systematic link between Schur-Weyl duality and higher transcendental invariants, revealing that higher diagonals encode geometric phenomena invisible to standard tautological classes.