Hodge theory of the Turaev cobracket and the Kashiwara–Vergne problem
Richard Hain
Duke University, Durham, USA
Abstract
In this paper we show that, after completing in the -adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve with a quasi-algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper on the Goldman bracket to construct torsors of solutions to the Kashiwara–Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these groups. Along the way, we give a homological description of the Turaev cobracket.
Cite this article
Richard Hain, Hodge theory of the Turaev cobracket and the Kashiwara–Vergne problem. J. Eur. Math. Soc. 23 (2021), no. 12, pp. 3889–3933
DOI 10.4171/JEMS/1088