Higher Symplectic Geometry

Understand

We consider generalizations of symplectic manifolds called n-plectic manifolds.

• A manifold is n-plectic if it is equipped with a closed, nondegenerate form of degree n+1.
• We show that higher structures arise on these manifolds which can be understood as the categorified or homotopy analogues of important structures studied in symplectic geometry and geometric quantization.
• Just as a symplectic manifold gives a Poisson algebra of functions, we show that any n-plectic manifold gives a Lie n-algebra containing certain differential forms which we call Hamiltonian.