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.