The Calabi homomorphism, Lagrangian paths and special Lagrangians

Let O be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold (X,ω) We define a functional C: O→ℝ for each differential form β of middle degree satisfying βΛω=0 and an exactness condition. If the exactness condition does not hold, C is defined on the universal cover of O. A particular instance of C recovers the Calabi homomorphism. If β is the imaginary part of a holomorphic volume form, the critical points of C are special Lagrangian submanifolds. We present evidence that C 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 C is convex on an open subspace O⁺⊂ O. As a prerequisite, we define a Riemannian metric on O⁺ 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.