The Calabi homomorphism, Lagrangian paths and special Lagrangians

Abstract: Let $\OO$ 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 $\CC:\OO \to \R$ for each differential form $\beta$ of middle degree satisfying $\beta \wedge \omega = 0$ and an exactness condition. If the exactness condition does not hold, $\CC$ is defined on the universal cover of $\OO.$ A particular instance of $\CC$ recovers the Calabi homomorphism. If $\beta$ is the imaginary part of a holomorphic volume form, the critical points of $\CC$ are special Lagrangian submanifolds. We present evidence that $\CC$ 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 $\CC$ is convex on an open subspace $\OO⁺ \subset \OO.$ As a prerequisite, we define a Riemannian metric on $\OO<sup>+$ 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.