Higher geometric quantization
In this talk, I will present some recent work in progress on geometrically quantizing "2-plectic manifolds" i.e. manifolds equipped with a closed non-degenerate 3-form. I will sketch how such a quantization procedure arises as a simple categorification of Sniatyki's "Bohr-Sommerfeld" quantization of a symplectic manifold equipped with a real polarization. In particular, just as quantizing a symplectic manifold gives a Hilbert space containing polarized sections of a line bundle, quantizing a 2-plectic manifold gives a module category over the symmetric monoidal category of finite dimensional Hilbert spaces. Moreover, this quantum category is built from polarized sections of a certain stack, or "2-line bundle". When the 2-plectic manifold is a compact simple Lie group equipped with the canonical Cartan 3-form, the resulting quantum state category is related to the representation category of the corresponding loop group.