The Calabi homomorphism, Lagrangian paths and special Lagrangians

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