Notes on mixed Teichm\"uller motives

TL;DR

This paper introduces mixed Teichmüller motives as motivic local systems on moduli spaces, establishing their categorical structure and connections to motivic correlators, extending concepts from elliptic motives to higher genus cases.

Contribution

It demonstrates that the category of mixed Teichmüller motives is equivalent to a subcategory of mixed Tate and elliptic motives, and links fundamental torsors to these motives.

Findings

Category of mixed Teichmüller motives is equivalent to a subcategory of mixed Tate and elliptic motives.

Unipotent fundamental torsors induce pro-objects in the mixed Teichmüller motives category.

Provides a realization of Goncharov's motivic correlators.

Abstract

As a higher genus version of universal mixed elliptic motives by Hain and Matsumoto, we consider mixed Teichm\"uller motives as certain motivic local systems on the moduli space of pointed curves. We show that the category of mixed Teichm\"uller motives is equivalent to a full subcategory of a certain product category of mixed Tate motives over Z and universal mixed elliptic motives. Furthermore, we show that unipotent fundamental torsors for the universal open curve give rise to a pro-object in the category of mixed Teichm\"uller motives. Our results can give a realization of motivic correlators proposed by Goncharov.