Drinfeld associators and Kashiwara-Vergne associators in higher genera

TL;DR

This paper constructs higher genus Kashiwara-Vergne associators from Gonzalez-Drinfeld associators, extending the genus 0 framework and analyzing framing implications, thus advancing the understanding of associators in surface topology.

Contribution

It introduces a method to derive higher genus Kashiwara-Vergne associators from Gonzalez-Drinfeld associators, generalizing previous genus 0 results.

Findings

Construction of genus g Kashiwara-Vergne associators from Gonzalez-Drinfeld associators

Proof based on Massuyeau's genus 0 work

Identification of specific framings in genus 1

Abstract

For $g\geq 0$, a genus $g$ Kashiwara-Vergne associator, introduced by Alekseev-Kawazumi-Kuno-Naef as a solution to the generalised KV equations in relation to the formality problem of the Goldman-Turaev Lie bialgebra on an oriented surface with a framing, is directly constructed from a genus $g$ analogue of a Drinfeld associator formulated by Gonzalez, which we call a Gonzalez-Drinfeld associator. The proof is based on Massuyeau's work in genus $0$. The framing is determined from the choice of a Gonzalez-Drinfeld associator, and in the case of genus $1$, we show that only particular framings are realised by our construction.