Deformations of Courant algebroids and Dirac structures via blended structures

Abstract

Deformations of a Courant algebroid (E,〈⋅,⋅〉,∘,ρ) and its Dirac subbundle A have been widely considered under the assumption that the pseudo-Euclidean metric 〈⋅,⋅〉 is fixed. In this paper, we attack the same problem in a setting that allows 〈⋅,⋅〉 to deform. Thanks to Roytenberg, a Courant algebroid is equivalent to a symplectic graded Q-manifold of degree 2. From this viewpoint, we extend the notions of graded Q-manifold, DGLA and L∞-algebra all to “blended” versions to combine two differentials of degree ±1 together, so that Poisson manifolds, Lie algebroids and Courant algebroids are unified as blended Q-manifolds; and define a submanifold A of “coisotropic type” which naturally generalizes the concepts of coisotropic submanifolds, Lie subalgebroids and Dirac subbundles. It turns out the deformations of a blended homological vector field Q are controlled by a blended DGLA, and the deformations of A are controlled by a blended L∞-algebra. The results apply to the deformations of a Courant algebroid and its Dirac structures, the deformations of a Poisson manifold and its coisotropic submanifold, and the deformations of a Lie algebroid and its Lie subalgebroid.