JournalsjemsVol. 23, No. 12pp. 3889–3933

Hodge theory of the Turaev cobracket and the Kashiwara–Vergne problem

  • Richard Hain

    Duke University, Durham, USA
Hodge theory of the Turaev cobracket and the Kashiwara–Vergne problem cover
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Abstract

In this paper we show that, after completing in the II-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve XX 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