4
$\begingroup$

See here. I present Theorem 6 and Corollary 7 as follows.

Theorem 6. For large $s$, $\Delta_s$ and $H$ have the same number of eigenvalues in the interval $[0, s^{-2/5}]$.

Corollary 7. $\dim \ker (\Delta_s) \le \# \text{ (eigenvalues of }H\text{ in }[0, s^{-2/5}]\text{).}$

It seems to me these two results are pulled out of nowhere in the text. Can someone explain to me the intuition behind them?

$\endgroup$
10
$\begingroup$

Why not go to Witten's actual paper? He explains it well, and here is a sketch:

The operator $\Delta_s$ depends on a Morse function $f:M\to\mathbb{R}$, and it takes on the form $$\Delta_s = \Delta+ s^2|df|^2 + s\cdot F$$ where $F$ is something depending on derivatives of $f$ (basically the Hessian). The point is that for $s>>0$ this is dominated by $|df|^2$, except at the critical points (where $df$ vanishes). So the collection of 'small' eigenvectors of $\Delta_s$ will be localized around those critical points (to compensate). We can thus approximate $\Delta_s$ near those critical points to analyze the eigenvalues (and that approximation is your "model operator" $H$). Your stated theorem is quantifying this approximation. Your stated corollary follows because those eigenvalues actually vanish for large $s$.

I forgot where I learned the rigorous approach, but to isolate around these critical points a partition of unity is used, where the size of the supports scales as $s^{2/5}$ I think.

There is "physics intuition" behind this approach (also explained in Witten's paper), but mathematically it is a restatement of the above. Roughly, the "Hamiltonian" $\Delta_s$ looks like the "potential energy" $s^2|df|^2$ for large $s$, and so this system is modeled by a bunch of "harmonic oscillators" for which we know the energy states.

$\endgroup$
  • $\begingroup$ which paper of Witten is this? I didn't see it in the link of OP, but, perhaps I just missed it. $\endgroup$ – James S. Cook Jul 30 '15 at 1:51
  • $\begingroup$ The paper is titled "Supersymmetry and Morse Theory", journal link here: projecteuclid.org/euclid.jdg/1214437492 although as of this writing it appears that you can also find PDF copies on google. $\endgroup$ – j.c. Jul 30 '15 at 12:04
6
$\begingroup$

$s^{-2/5}$ comes from finding an optimal estimate of the middle term in the Witten Laplacian in terms of the Laplacian and the potential. In order to understand this, you need to look up the actual derivations of the estimates.

Here are couple of good places where you can find these estimates:

  1. Schrödinger Operators: With Applications to Quantum Mechanics and Global Geometry (Theoretical and Mathematical Physics), by Cycon, Froese, ...
  2. M. Shubin, Novikov inequalities for vector fields, in The Gelfand mathematical Seminars 1993-1995, Burkhauser 1996.

My understanding of Witten's proof is as follows. One of the cruxes of his proof is the Hodge isomorphism, which acts as a bridge between topology and functional analysis. The number of obstructions to moving submanifolds around in a given manifold is captured by the Betti numbers (coming from de Rham cohomology), and the Hodge isomorphism says that these are equal to the nullity of a differential operator, namely the Laplacian$$\Delta:= \text{d}\text{d}^\dagger + \text{d}^\dagger \text{d},$$restricted to act on certain subspaces, namely forms of a given rank. This is undoubtedly a powerful result to have handy since, like its far more general cousin, the Atiyah-Singer index theorem, it allows us to probe global properties of a manifold using local objects (local calculations, rather), which are far easier to handle.

The Morse inequalities, as we know, place lower bounds on the number of critical points with a given Morse index a function defined on a manifold can have using the Betti numbers of the manifold. There are complicated ways of establishing these using CW complexes, but Witten came up with a far more straightfoward demonstration. The idea is to replace the exterior derivative $\text{d}$ by$$\text{d}_t = e^{ht} \text{d}e^{-ht},$$and consequently, $d^\dagger$ by$$d^\dagger_t = e^{-ht}d^\dagger e^{ht},$$where $h$ is the function whose critical points we are interested in, and $t$ is a real number. Note that when we take the adjoint of the composition of operators, we reverse the order in which their individual adjoints act, which is precisely what we have done here.

Since $\text{d}_t$ is related to $\text{d}$ by a similarity transformation, the kernels (respectively images) of $\text{d}_t$ is isomorphic to the kernels (respectively images) of $\text{d}$ under the invertible map$$\omega \mapsto e^{ht}\omega.$$This means that the Betti numbers are the same. However, the Laplacians$$\Delta_t := \text{d}_t\text{d}_t^\dagger+ \text{d}_t^\dagger\text{d}_t$$and $\Delta$ are not related by a similarity transformation, so there is no nice way in which the kernel of $\Delta_t$ is related to the kernel of $\Delta$. However, the proof of the Hodge isomorphism carries over to the case of $\Delta_t$ without any trouble, and it follows that the Betti numbers are dimensions of the kernels of $\Delta_t$ restricted to act on forms of a certain rank.

Now, the spectrum of $\Delta_t$ simplifies considerably in the limit $t \to \infty$ because when we expand $\Delta_t$, there is a term quadratic in $t$ whose coefficient is $|\text{d}h|^2$. This means that the eigenvalues corresponding to an eigenform of $\Delta_t$ grows like $t^2$ unless the form is localized around precisely those points where $\text{d}h = 0$. In other words, the eigenforms belonging to zero become localized around the critical points of the function $h$ as $t$ becomes large. By doing a local analysis around these critical points, we may show that a $p$-form which is harmonic with respect to $\Delta_t$ can localize only around critical points of Morse index $p$ (which is basically the number of negative eigenvalues of the Hessian at the point, or in other words, the number of independent directions in which the function $h$ decreases as we move away from the critical point, assuming of course that the Morse function is nondegenerate). The Morse inequalities follow naturally from this observation.


We augment the above with some philosophical reflection. Mathematically, the proof could be considered "not that hard" on some level; Witten is just abusing deformations of the Laplacian and showing that this gives us Morse theory and more. Having Morse cohomology for infinite-dimensional spaces (and getting the right notion of infinite-dimensional geometry in general) is hard. However, Witten is interpreting notions in a way that is very amenable to generalization to the infinite-dimensional case; he has basically given an explicit complex for cohomology that is very "finite-dimensional natured," i.e. very amenable to allowing us to define a cohomology theory for say, a space of maps. This gives us something along the lines of Floer cohomology.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy