The Calabi homomorphism, Lagrangian paths and special Lagrangians

Abstract

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.