Harmonic Magnus Expansion on the Universal Family of Riemann Surfaces

TL;DR

This paper introduces the harmonic Magnus expansion, a higher analogue of the period matrix for Riemann surfaces, which induces a flat connection capturing higher Johnson homomorphisms and representing Morita-Mumford classes.

Contribution

It develops the harmonic Magnus expansion using Chen's iterated integrals, linking it to the mapping class group and Morita-Mumford classes on the moduli space.

Findings

Defines the harmonic Magnus expansion as a higher period matrix.

Constructs a flat connection encoding higher Johnson homomorphisms.

Provides explicit differential forms representing Morita-Mumford classes.

Abstract

Let ${\mathbb M}_{g, 1}$, $g \geq 1$, be the moduli space of triples $(C, P_0, v)$ of genus $g$, where $C$ is a compact Riemann surface of genus $g$, $P_0 \in C$, and $v \in T_{P_0}C\setminus\{0\}$. Using Chen's iterated integrals we introduce a higher analogue of the period matrix for a triple $(C, P_0, v)$, {\it the harmonic Magnus expansion}. It induces a flat connection on a vector bundle over the space ${\mathbb M}_{g, 1}$, whose holonomy gives all the higher Johnson homomorphisms of the mapping class group. The connection form, which is computed as an explicit quadratic differential, induces "canonical" differential forms representing (twisted) Morita-Mumford classes and their higher relators on ${\mathbb M}_{g, 1}$. In particular, we construct a family of twisted differential forms on ${\mathbb M}_{g, 1}$ representing the $(0, p+2)$-twisted Morita-Mumford class $m_{0, p+2}$ combinatorially parametrized by the Stasheff associahedron $K_{p+1}$.