UC Riverside

In higher symplectic geometry, we consider generalizations of symplectic manifolds called n-plectic manifolds. We say a manifold is n-plectic if it is equipped with a closed, non-degenerate form of degree (n+1). We show that certain higher algebraic and geometric structures naturally arise on these manifolds. These structures can be understood as the categorified or homotopy analogues of important structures studied in symplectic geometry and geometric quantization. Our results imply that higher symplectic geometry is closely related to several areas of current interest including string theory, loop groups, and generalized geometry.

We begin by showing that, just as a symplectic manifold gives a Poisson algebra of functions, any n-plectic manifold gives a Lie n-algebra containing certain differential forms which we call Hamiltonian. Lie n-algebras are examples of strongly homotopy Lie algebras. They consist of an n-term chain complex equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity.

We then develop the machinery necessary to geometrically quantize n-plectic manifolds. In particular, just as a prequantized symplectic manifold is equipped with a principal U(1)-bundle with connection, we show that a prequantized 2-plectic manifold is equipped with a U(1)-gerbe with 2-connection. A gerbe is a categorified sheaf, or stack, which generalizes the notion of a principal bundle. Furthermore, over any 2-plectic manifold there is a vector bundle equipped with extra structure called a Courant algebroid. This bundle is the 2-plectic analogue of the Atiyah algebroid over a prequantized symplectic manifold. Its space of global sections also forms a Lie 2-algebra. We use this Lie 2-algebra to prequantize the Lie 2-algebra of Hamiltonian forms.

Finally, we introduce the 2-plectic analogue of the Bohr-Sommerfeld variety associated to a real polarization, and use this to geometrically quantize 2-plectic manifolds. For symplectic manifolds, the output from quantization is a Hilbert space of quantum states. Similarly, quantizing a 2-plectic manifold gives a category of quantum states. We consider a particular example in which the objects of this category can be identified with representations of the Lie group SU(2).