# Integral of symplectic form?

It is often said that "differential forms are used for integration".

Typically people like to talk about the integral $$\int_M \omega$$ of a differential form $$\omega$$, and exterior derivative, one of the most important operation of differential form, is defined such that Stokes' theorem $$\int_{\partial M} \omega = \int_M \mathrm{d} \omega$$ can hold.

Symplectic manifold is a manifold equipped with a symplectic 2-form $$\omega$$. Therefore, according to the former consideration, it should be interesting to talk about integral of the symplectic form on a two-dimensional submanifold of a symplectic manifold.

However, it seems that people in the field of symplectic geometry is quite indifferent on this. Why?

This is only true if you are very pedantic about what you mean by "symplectic geometry". Namely, if you include symplectic topology, then one of the most important invariants of a symplectic manifold requires integrating the symplectic form over $$2$$-dimensional submanifolds. To be more precise, one considers maps $$u:\Sigma\to M$$ for $$\Sigma$$ a Riemann surface, and then integrates $$\int_\Sigma u^*\omega$$. If $$u$$ is an embedding, then this is clearly the same as integrating over the submanifold $$u(\Sigma)\subseteq M$$.

The reason why this is done, is because one wants to "count" the number of pseudo-holomorphic maps $$\Sigma\to M$$, up to reparameterisation. To be able to do this, one has to restrict to maps with finite energy, i.e. $$\int_\Sigma u^*\omega<\infty$$. Then the "moduli space" of solutions can be constructed, and compactified (finite energy is crucial to ensure compactification). If this moduli space has expected dimension $$0$$, then the compactification results in a finite number of points, which can be counted in some appropriate sense. The numbers which are obtained in this way are called the Gromov-Witten invariants of the symplectic manifold.

While I love the answer of Quaere Verum and cant give enough +1s on it, here is a few more "simple" facts about the symplectic form $$\omega$$ on a symplectic manifold $$(M,\omega)$$ of dimension $$2n$$:

First of all, note that a $$2$$-form on a $$2n$$ dimensional manifold (unless $$n=1$$) is not really meant to be integrated. But let me explain why closed stll matters and how we can still integrate using this fact:

$$\omega^n$$ is a volume form, called the symplectiv volume form. It gives you a "canonical" volume form on your manifold compatible with the structure. If in addition $$M$$ is closed (i.e. compact with no boundary), then you know that all the even cohomology groups $$H^{2k}(M)\neq 0$$. Use the previous fact to proof this!

Hamiltonian vector fields, moment maps - they all make use of the symplectic form. It usually involves the Cartan formula, which for the symplectic form just reads $$\mathcal{L}_X\omega=d(i_{X}\omega)$$, where $$d$$ is exeterior differentiation and $$i$$ is the contraction/interior product with a vector field $$X$$.

Last but not least, we also have so called Kähler manifolds, where the symplectic form can be seen as an "intermediate" form between the Riemannian metric $$g$$ and the (almost-)complex structure $$J$$ on M, i.e. we have $$\omega(\cdot,\cdot)=g(\cdot,J\cdot).$$ This construction would define you a non-degenerate $$2$$-form for any choice of $$J$$ and $$g$$, however, it is not necessarily closed.