2-Plectic geometry, Courant algebroids, and categorified prequantization

Abstract

A 2-plectic manifold is a manifold equipped with a closed nondegenerate 3-form, just as a symplectic manifold is equipped with a closed nondegenerate 2-form. In 2-plectic geometry one finds the higher analogues of many structures familiar from symplectic geometry. For example, any 2-plectic manifold has a Lie 2-algebra consisting of smooth functions and Hamiltonian 1-forms. This is equipped with a Poisson-like bracket which only satisfies the Jacobi identity up to “coherent chain homotopy”. Over any 2-plectic manifold is a vector bundle equipped with extra structure called an exact Courant algebroid. This Courant algebroid is the 2-plectic analogue of a transitive Lie algebroid over a symplectic manifold. Its space of global sections also forms a Lie 2-algebra. We show that this Lie 2-algebra contains an important sub-Lie 2-algebra which is isomorphic to the Lie 2-algebra of Hamiltonian 1-forms. Furthermore, we prove that it is quasi-isomorphic to a central extension of the (trivial) Lie 2-algebra of Hamiltonian vector fields, and therefore is the higher analogue of the well-known Kostant–Souriau central extension in symplectic geometry. We interpret all of these results within the context of a categorified prequantization procedure for 2-plectic manifolds. In doing so, we describe how $U(1)$-gerbes, equipped with a connection and curving, and Courant algebroids are the 2-plectic analogues of principal $U(1)$ bundles equipped with a connection and their associated Atiyah Lie algebroids.