The Calabi homomorphism, Lagrangian paths and special Lagrangians

Let be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold We define a functional for each differential form of middle degree satisfying and an exactness condition. If the exactness condition does not hold, is defined on the universal cover of A particular instance of recovers the Calabi homomorphism. If is the imaginary part of a holomorphic volume form, the critical points of are special Lagrangian submanifolds. We present evidence that is related by mirror symmetry to a functional introduced by Donaldson to study Einstein-Hermitian metrics on holomorphic vector bundles. In particular, we show that is convex on an open subspace As a prerequisite, we define a Riemannian metric on and analyze its geodesics. Finally, we discuss a generalization of the flux homomorphism to the space of Lagrangian submanifolds, and a Lagrangian analog of the flux conjecture.