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 2algebra. We show that this Lie 2-algebra contains an important sub-Lie 2-algebra which is isomorphic to the Lie 2-algebra of Hamiltonian 1forms. 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 2plectic 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.

Generalized Higher Gauge Theory and M5-brane dynamics

“…Both were seen to be examples of specific n-algebras. We will see now that they are in fact related, since the latter contain the structure of the former, as was shown in 40 .…”

$L_{\infty}$-algebras of local observables from higher prequantum bundles

“…The latter is the Lie algebra of global sections of the Atiyah algebroid of P (see, for example, Sec. 2 of [33] and Def. 5.1.1 below).…”

Extended Riemannian Geometry I: Local Double Field Theory

“…is to be seen as the Atiyah algebroid of the gerbe with Dixmier-Douady class H, see also [63]. The infinitesimal symmetries 8 of a U(1)-gerbe with characteristic class H over M are described locally over patches U i by the exact Courant algebroid V 2 (U i ) with Ševera class H. The 1-forms Λ ij introduced above, which describe the connection on the gerbe can now be used to glue together the local descriptions T U i ⊕ T * U i over overlaps of patches U i ∩ U j by mapping…”