The Calabi homomorphism, Lagrangian paths and special Lagrangians

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