Towards higher geometric quantization

Manifolds equipped with a closed non-degenerate form of degree > 2 can be thought of as higher analogs of symplectic manifolds. In this talk, I will describe some work in progress on developing a geometric quantization procedure for such manifolds. In particular, the 3-form case involves structures such as U(1)-gerbes, twisted vector bundles, and linear categories; just as ordinary geometric quantization uses U(1)-bundles, global sections of line bundles, and Hilbert spaces. Comparisons will be made with related subjects, such as Dirac geometry, and applications to representation theory will be discussed.