Energy quantization for $J$-holomorphic spheres

Question

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.

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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?

Answer 1

There is indeed something that links J-holomorphic curves, harmonic forms and energy eigenstates in quantum mechanics: a variational principle. That is to say, they can all be characterized by the extremization of a certain functional. This offers an intuitive understanding of their quantized behaviour: if you draw a random curve on a sheet of paper, the local minima will typically be a discrete set. In fact I would say the variational principle is a deeper link than the quantization, but let me first get into more detail:

1. A J-holomorphic curve is usually of the form $u: (\Sigma_g,j) \to (M,J,\omega)$ with the domain a Riemann surface, and the codomain a symplectic manifold with an $\omega$-tame $J$, such that $u$ is J-holomorphic (i.e. $\bar \partial_J (u) := \mathrm d u + J \circ \mathrm d u \circ j = 0$). Let us for the moment allow general smooth maps $u$, then it's straight-forward to show that if $J$ is in fact $\omega$-compatible:
   
   $\qquad \qquad \qquad \boxed{E(u)} := \frac{1}{2} \int |\mathrm d u|^2 \; \mathrm d \textrm{vol} = \boxed{\int |\bar \partial_J u|^2 \; \mathrm d \textrm{vol} + \int u^* \omega}$
   
   Since J-holomorphic curves are exactly those with $\bar \partial_J u = 0$, we see that they extremize this energy functional, leaving the usual topological term $\int u^* \omega$.

2. Another analytic object that gives rise to quantization are harmonic forms: for example Hodge theory tells us (when) they represent the cohomology of our manifold (or relatedly there's the quantization of harmonic maps, as you mention). To understand them from a variational perspective: suppose we have a de Rham cohomology class $[\alpha] \in H^k(M)$ and let's look for the representative that minimizes the $\int g(\cdot, \cdot)$-norm. It should be such that for all $\beta \in \Omega^{k-1}(M)$ we have
   
   $\qquad \qquad \qquad \frac{\mathrm d}{\mathrm dt} \langle \alpha + t\; \mathrm d \beta, \alpha + t \;\mathrm d \beta \rangle \big|_{t=0} = 0$
   
   I.e. we want $\langle \alpha , \mathrm d \beta \rangle = \langle \delta \alpha, \beta \rangle = 0$ for all $\beta$, which means $\boxed{\delta \alpha = 0}$. Since $\alpha$ is a cohomology representative, it's closed $\boxed{\mathrm d \alpha = 0}$. These conditions are equivalent to being harmonic: $\boxed{\Delta \alpha = 0}$. The harmonic representative of a cohomology class is the one that extremizes the norm!

3. Finally, one formulation of quantum mechanics is in terms of path integrals. Classical point particles correspond to extrema of the classical action, but in fact there's a variational principle for the quantum wave functions such that the energy wavefunctions come out. I might have to come back at a later time to flesh this out.

So a variational principle is present. The reason I would give this more importance than the quantization, is because the latter depends on our boundary conditions. For example, if our manifold is not compact, then there is no quantization for $E(u)$, or no Hodge theory. Or similarly if instead of a quantum wave function in a potential well we have one in free space, energy won't be quantized. Yet even in these cases, the variational principle can still make sense.

But is this an answer? Or are we left with the question:

Why should the J-holomorphic curve (et cetera) be characterized by a variational principle?

To me this seems like a very interesting question to which I sadly have no answer!