Liouville class of a Lagrangian submanifold ($[\lambda {|}_L]$)

For a Lagrangian submanifold $L\subset ({\mathbb{R}}^{2n},d\lambda )$, the restriction $\lambda {|}_L$ is closed. Its cohomology class is preserved by ambient symplectomorphisms because every closed one-form on ${\mathbb{R}}^{2n}$ is exact. On a product torus of circles, its periods are the enclosed symplectic areas of the factors.

For a smooth nonconstant holomorphic disk $h$ in ${\mathbb{C}}^n$ with boundary on a Lagrangian submanifold $L$, its symplectic area is ${\int }_{D^2}|{\partial }_{s}h{|}^{2}dsdt>0$. The Cauchy-Riemann equations give ${\partial }_{t}h=J{\partial }_{s}h$, so this is the pullback of the standard symplectic form. The Stokes theorem identifies the area with the period of a primitive along the boundary loop. Thus the Liouville class of a Lagrangian submanifold evaluates nontrivially on this loop and cannot vanish.

For a Lagrangian submanifold $L$ in an exact symplectic manifold with primitive $\lambda$, its period subgroup is $\{{\int }_c\lambda :c\in H_1(L;\mathbb{Z})\}\subset \mathbb{R}$. It is rational when this subgroup is discrete. If it is nonzero, it is $\gamma \mathbb{Z}$ for a unique $\gamma >0$. Some conventions reserve the word rational for this nonzero case. The zero subgroup corresponds to an exact differential form $\lambda {|}_L$; its least positive period is interpreted as $+\infty$ if that endpoint convention is admitted.

For a nonzero discrete Liouville period group, this is its positive generator. In Euclidean symplectic space, the same group is obtained from signed symplectic areas of disks with boundary on $L$, by the Stokes theorem and contractibility of the ambient space. A product of circles of common radius $r$ is a rational Lagrangian submanifold with $\gamma (L)=\pi r^2$. Translation changes a primitive only by an exact form and hence leaves the period group unchanged.