Cartier integration of infinitesimal 2-braidings via 2-holonomy of the CMKZ 2-connection, I: Hexagonators and the Breen polytope

TL;DR

This paper constructs a candidate hexagonator series using 2-holonomy of a 2-connection over the configuration space of particles, linking it to infinitesimal 2-braidings and the Breen polytope axiom.

Contribution

It introduces a novel approach to integrating infinitesimal 2-braidings via 2-holonomy and verifies the Breen polytope axiom within this framework.

Findings

Second-order hexagonator term matches previous infinitesimal results.

Constructed a candidate hexagonator series using 2-holonomy.

Proved the Breen polytope axiom holds under certain conditions.

Abstract

This paper follows on from "Syllepses from 3-shifted Poisson structures and second-order integration of infinitesimal 2-braidings". Given a symmetric strict infinitesimal 2-braiding on a symmetric strict monoidal cochain 2-category, we construct a candidate hexagonator series as a 2-holonomy with respect to the Cirio-Martins-Knizhnik-Zamolodchikov (CMKZ) fake flat 2-connection over the configuration space of 3 distinguishable particles on the complex line, $Y_3$. The second-order term of the hexagonator series is computed and found to agree with the "infinitesimal hexagonator" from the aforementioned work. Finally, we assume a coherent totally symmetric strict infinitesimal 2-braiding and prove that the Breen polytope axiom is satisfied by translating it into a 2-loop in $Y_3$.