Courant algebroids from categorified symplectic geometry

Abstract

In categorified symplectic geometry, one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed nondegenerate (n+1)-form. The case relevant to classical string theory is when n=2 and is called "2-plectic geometry". Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, there is a Lie 2-algebra of observables associated to any 2-plectic manifold. String theory, closed 3-forms and Lie 2-algebras also play important roles in the theory of Courant algebroids. Courant algebroids are vector bundles which generalize the structures found in tangent bundles and quadratic Lie algebras. It is known that a particular kind of Courant algebroid (called an exact Courant algebroid) naturally arises in string theory, and that such an algebroid is classified up to isomorphism by a closed 3-form on the base space, which then induces a Lie 2-algebra structure on the space of global sections. In this paper we begin to establish precise connections between 2-plectic manifolds and Courant algebroids. We prove that any manifold M equipped with a 2-plectic form omega gives an exact Courant algebroid E_omega over M with Severa class [omega], and we construct an embedding of the Lie 2-algebra of observables into the Lie 2-algebra of sections of E_omega. We then show that this embedding identifies the observables as particular infinitesimal symmetries of E_omega which preserve the 2-plectic structure on M.Comment: These preliminary results have been superseded by those given in arXiv:1009.297