A universal version of the KZB flat bundle and extensions of the Grothendieck–Teichmüller Lie algebra

We construct a bundle with flat connection over the moduli space of marked curves of genus one; this is a universal version of the KZB connection on the same moduli space. We study the holonomy representation of this connection. The building blocks of this representation are analytic objects, satisfying genus one analogues of the associator identities. The corresponding scheme is a torsor under an extension of the Grothendieck–Teichmüller Lie algebra, which we make explicit. We discuss the analogous constructions in genus greater that one. (Partly joint with D. Calaque and P. Etingof).