Deformation theory refers to an apparatus in many parts of math and physics for going from an infinitesimal (= first order) deformation to a full deformation, either formal or convergent appropriately. If the algebra being deformed is that of observables, the result is deformation quantization, independent of any realization in terms of Hilbert space operators. There are very important but rare cases in which a formula for a full deformation is known. For physics, the most important is the Moyal-Weyl star product formula. In this paper, we concentrate on deformations of Courant algebroid structures via star products on graded manifolds. In particular, we construct a graded version of the Moyal-Weyl star product. Recently, in Double Field Theory (DFT), deformations of the C-bracket and O(d,d)-invariant bilinear form to first order in the closed string sigma model coupling α′ were derived by analyzing the transformation properties of the Neveu-Schwarz B-field. By choosing a particular Poisson structure on the Drinfel’d double corresponding to the Courant algebroid structure of the generalized tangent bundle, we reproduce these deformations for a specific solution of the strong constraint of DFT as expansion of a graded version of the Moyal-Weyl star product.

# Star products on graded manifolds and α’-corrections to Courant algebroids from string theory

J. Math. Phys. 56 (2015) 9, 092302