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Let $(\mathbb{CP}^1, j, g_{\text{FS}})$ be the complex projective line with the standard complex structure and the Fubini-Study metric and let $(M,J,\omega,g)$ be an almost Kähler compact manifold ($\omega$ is closed, $J$ is not neccesarily integrable). Using energy estimates, it can be shown that there is a constant $\hbar > 0$ that depends on $J$ and $g$ (and $\omega$) such that any non-constant $J$-holomorphic sphere $u : \mathbb{CP}^1 \rightarrow M$ satisfies $$ E(u) = \int_{\mathbb{CP}^1} |du|^2 \mathrm{dvol}_{\text{FS}} \geq \hbar. $$

That is, a non-constant $J$-holomorphic sphere must have an energy of at least $\hbar$. A more elaborate analysis can show that the set of possible "energy levels" $$ \{ E(u) \}_{u \; \text{is } J-\text{holomorphic}} $$ is a discrete set. This phenomenon also occurs for two-dimensional harmonic maps and presumably in other settings which I'm less familiar with.

Now, this smells in some sense like a "quantization". Smooth spheres $u: \mathbb{CP}^1 \rightarrow M$ are much less rigid with respect to energy - we can pertube them a little, changing the energy a little. However, once we impose some conditions on the maps (being $J$-holomorphic, harmonic), we get a discrete "spectrum", finite dimensional moduli spaces, etc.

Is there any physical interpretation behind this phenomenon? Any hueristic that explains intuitively why this should happen (maybe as some sort of a "quantization")? Is there a theory in which the set of possible energy levels have some physical meaning, analogous to the a spectrum of some self-adjoint operator in quantum mechanics? Can one think of the set of possible energy levels as the "spectrum" of the non-linear $\bar{\partial}_J$ and learn interesting things on the (almost complex) geometry of $M$ from it?

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The energy depends only on the homology class of the curve, which is an element of a discrete group, whence the name quantization attached to this phenomenon. The fact that the target manifold is symplectic plays a role in this. –  Liviu Nicolaescu Dec 16 '12 at 21:44
Yeah, but a priori this fact does not immediately imply that the possible energy levels are discrete for the image of the pairing $A \mapsto \left<\omega, A\right>$ from $H_2(M;\mathbb{Z}) \rightarrow \mathbb{R}$ might be non-discrete. In the first edition of McDuff-Salamon, the existence of $\hbar$ is deduced from a non-trivial analytical argument (estimate of the $C^{\infty}$ norm of the differential in terms of energy, under the assumption of small energy). However, I just found out that in the new edition there is a simple argument of a few lines that requires no additional analysis, so in –  levap Dec 16 '12 at 22:28
this sense, one really doesn't need anything "deeper" than the fact that $\omega$ is closed. –  levap Dec 16 '12 at 22:28
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