Goldman-Turaev formality from the Kontsevitch integral
Quantum Algebra
2025-09-30 v2
Abstract
We present a new solution to the formality problem for the framed Goldman--Turaev Lie bialgebra, constructing Goldman-Turaev homomorphic expansions (formality isomorphisms) from the Kontsevich integral. Our proof uses a three dimensional derivation of the Goldman-Turaev Lie biaglebra arising from a low-degree Vassiliev quotient -- the {\em emergent} quotient -- of tangles in a thickened punctured disk, modulo a Conway skein relation. This is in contrast to Massuyeau's 2018 proof using braids. A feature of our approach is a general conceptual framework which is applied to prove the compatibility of the homomorphic expansion with both the Goldman bracket and the technically challenging Turaev cobracket.
Cite
@article{arxiv.2509.20983,
title = {Goldman-Turaev formality from the Kontsevitch integral},
author = {Dror Bar-Natan and Zsuzsanna Dancso and Tamara Hogan and Jessica Liu and Nancy Scherich},
journal= {arXiv preprint arXiv:2509.20983},
year = {2025}
}