Star products on graded manifolds and $\alpha'$-corrections to double field theory

Abstract

Originally proposed as an $O(d,d)$-invariant formulation of classical closed string theory, double field theory (DFT) offers a rich source of mathematical structures. Most prominently, its gauge algebra is determined by the so-called C-bracket, a generalization of the Courant bracket of generalized geometry, in the sense that it reduces to the latter by restricting the theory to solutions of a "strong constraint". Recently, infinitesimal deformations of these structures in the string sigma model coupling $\alpha'$ were found. In this short contribution, we review constructing the Drinfel'd double of a Lie bialgebroid and offer how this can be applied to reproduce the C-bracket of DFT in terms of Poisson brackets. As a consequence, we are able to explain the $\alpha'$-deformations via a graded version of the Moyal-Weyl product in a class of examples. We conclude with comments on the relation between $B$- and $\beta$-transformations in generalized geometry and the Atiyah algebra on the Drinfel'd double.