⇤Math diary
Jul 16, 2023: Witten's morse theory
In the last few weeks, I found myself a new little obsession: Witten’s holomorphic morse inequalities. It all started when I needed to compute the Dolbeaut cohomology groups $H^{p,q}$ of a particular manifold. And unfortunately, I have no idea how to do that. For ordinary cohomology, I know like 4 or 5 different intuitions and computation methods. All of these methods follow the same intuitive beats
- cut up the manifold into simple parts, whose cohomology we understand
- Figure out how cohomology changes when we glue things together
- Glue the parts back into the full manifold, and find its cohomology This process is tractable because topology is soooo floppy. There just aren’t very many ways to glue two things together! The gluing instructions are fundamentally combinatorial, so we can get computers or (in a pinch) undergrads to compute cohomology.
Unfortunately, Complex manifolds are not made of play-doh. They have a texture, a subtle “complex structure”, which has to match when gluing. Keeping track of the final texture, there are many ways to glue things, and I have no clue how to represent them combinatorially. I don’t feel complex structures in my bones. But there is one approach to cohomology that’s very similar in for complex and real manifolds, and luckily its my favorite one. In fancy words, it’s a combination of hodge theory and morse theory. Let me break it down:
Some analytic approaches to cohomology
Cohomology measures the topology of a manifold. We can approach it analytically, seeing the topology manifest from the properties of functions on the manifold.
A “Good morning” to morse theory
“Morse theory” studies topology through the structure of a function on a manifold. More intuitivly, I think of it as “topology through topography”. Think about a mountinous island. The island’s structure is represented by the height at each point, mathematically a function $f: \mathbb{R}^2 \to \mathbb{R}$. To represent this on a map, we draw the contours of the height function, which encodes the structure of the island (see picture below). Topologically, we need even less information. Imagine global warming strikes, and the ocean levels rise and rise until the island is fully submerged. At the beginning we have one island with two peaks, until eventually the ocean crosses the path between the peaks, and we have two separated islands. Rise some more, and the right peak submerges. Even more, and the left peak is claimed by the ocean, leaving nothing. If we ignore any details about the shapes of the islands, we can represent this progression as \(● \to ● \, ● \to ● \to \_\)
This simple diagram is all we need to reconstruct the two-peaked structure of the island. The changes in topology happen exactly where the land is flat, a critical point of the function. When this is a saddle point, the raising waters split an island into two. When it is a local maximum, the waters submerge one of the islands entirely. If we had a local minimum, like a depression filling with water, it would have made a hole in an island.
![[Pasted image 20230716114153.png]]
The point is, the topology of a submerging island changes in very predictable ways, exactly at the critical points of the function $f$. The type of change is controlled by the type of critical point, classified according to the second derivative of $f$. In particular, the number of directions with positive second derivative (the index of the critical point) contains this information. This leads us to the defining philosophy of morse theory:
[!thm] Morse theory The topology of a manifold $M$ is controlled by the structure of the critical points of a sufficiently nice function $f:M\to \mathbb{R}$
[!thm]- Technical details By “nice function”, we mean a “morse function”. First off, we need $f$ to be second differentiable, so at every critical point, $Df = 0$, we have a symmetric matrix of second derivatives $D^2 f$. $f$ is morse iff it is $C^2$ and the matrix $D^2 f$ is nondegenerate, meaning all eigenvalues are either positive or negative.
To fit this into the intuitive roadmap for cohomology calculations described above, we want to decompose the manifold $M$ according to the critical points of $f$. Following the topographic analogy, we consider watersheds. Think about the contentential united states. Imagine we drop a single drop of water somewhere, and follow its journey all the way until it ends up at an ocean. West of the rocky mountains, it ends up in the pacific. To the east, it will find its way to the Atlantic. We thus decompose the entire continent into two watersheds, delineated based on the final resting place of a drop of water, the pacific watershed and the atlantic one.
Let’s adapt this idea mathematically. suppose we have a manifold $M$ with a height function $f$. We imagine a drop of water (represented by a point $x$) will move along $M$ according to the gradient flow of $f$. That is, it traces a path $x(t)$ such that at any time $t$, the velocity is $\cdot x(t) = \nabla f(x(t))$. If $M$ is compact, the drop will always end up at some fixed point $p$. We define the watershed $W_p$ to be the set of starting points that end up at $p$ (this is usually called the stable manifold). Symbolically, $W_p$ is the set of $x\in M$ such that $\lim_{t\to \infty} x(t) = p$.
Think again about a dimension two example: Every mountain peak $p$ (a local maximum of $f$) has a watershed consisting only of $p$, as a drop balanced on the peak will not move. A ridge passing between two mountains (a saddle point of $f$ in dimension two) has a one dimensional watershed, a line connecting the two peaks of adjacent mountains to the saddle point. A local minimum acts like one of the oceans of America, with a two dimensional watershed. In general, the dimension of the watershed $W_p$ is the number of positive directions at the critical point $p$, its index. Furthermore, each watershed $W_p$ is topologically a disc, since it’s contractible.
Now we can compute cohomology of a manifold $M$. Choose a sufficiently nice height function $f$, and decompose $M$ into watersheds for each critical point of $f$. Symbolically, letting $C$ be the set of critical points of $f$, \(M = \bigcup\limits_{p\in C} W_p\) Since $W_p$ is just a disc, it has trivial topology. Every hole of dimension $d$ is necessarily composed of at least one $d$-dimensional watershed. So, this automatically gives us a bound: The cohomology of rank $d$, which counts the number of dimension $d$ holes, is less than the number of critical points of index $d$ of $f$! This is true for any possible nice function $f$ which you choose.
Now we need to glue the watersheds together. For example, a one dimensional watershed representing a pass between mountains glues to the watersheds at the peaks of both mountains. Similarly, a two dimensional watershed glues to a one dimensional watershed representing the mountain pass at the border. This, in turn, glues to each of the peaks.
In general, we see that the gluing is represented by a set of inclusions. critical points $p$ glues to $q$ if the watershed $W_q$ forms part of the border of $W_p$ (technically, if $W_q$ is a subset of the closure of $W_p$). This gives a partial ordering on the critical points. To compute the cohomology, each watershed $W_p$ potentially forms a hole of dimension the index of $p$ (i.e a cohomology class). However, there is some redundancy, and some critical points which don’t form any holes. We determine these from the gluing relations. In particular, the redundancy comes from counting flow lines between critical points of index differing by 1. This measures the number of possible directions (with sign) a water drop can take to travel from one critical point to a lower one with watershed of one greater dimension. With a bit of finagling (this part can get technically painful), we can use this to count holes, and compute cohomology!
[!thm]- Technical details For this to work, we need some transversality assumptions. Essentially, these ensure that the dimension of flowlines between $p$ and $q$ is indeed the difference in indicies of $p$ and $q$, minus 1. Such functions are called morse-smale.
Counting the flow lines with sign is a huge pain. We need some consistent way to assign a sign to each flow line, which leaves you in a soup of miserable sign errors. Either you can tough it out, or throw up your hands and compute cohomology in $\mathbb{Z}_2$ coefficients where signs dont matter. Because of these technical issues, finite dimensional morse homology is rarely used. Instead, this is a roadmap for an infinite dimensional version called floer homology, where the tecnhical pains are worth it.
Also, morse theory really gives a homology theory, not a cohomology one. Sorry :(
To summarize: Morse theory studies the topology of a manifold by decomposing it according to a nice function. The topology is encoded entirely in the structure of the critical points of the function. In particular, the number of critical points of each index bounds the number of holes of each dimension. With more care, we can “glue” the critical points together by counting flow lines, and exactly figure out the cohomology.
A “Hello” to hodge theory
Alas, morse theory as constructed above doesn’t help much in the complex case, as there is no clear way to glue to the watersheds together. But dolbeaut cohomology looks similar to de rahm cohomology in a different, more analytic perspective. The idea is, we can represent the cohomology as the space of solutions to a differential equation.
Cohomology classes are represented by closed differential forms, modulo addition by exact forms. We can pick a unique representative of any cohomology class by minimizing the integrated norm (or “energy”) among all closed forms in that class. But things that maximize or minimize a functional always satisfy some differential equation, given in general form as the euler lagrange equations. In this case, minimal energy forms have laplacian zero. Said another way, we can reduce energy of a differential form by smoothing it out, replacing the value at each point with the average nearby value. This is like letting heat diffuse across a manifold. The resulting form, stretched taught, has laplacian zero. Such a form is called harmonic.
[!thm] Hodge theory A differential form $\omega$ is said to be harmonic if $\Delta \omega = \text{d}^\text{d} +\text{d}\text{d}^ \omega = 0$ .
The cohomology groups $H^d(M,\mathbb{R})$ are isomorphic to the vector space of harmonic $d$-forms, $\mathcal{H}^d(M) = \ker (\Delta : \Omega^d \to \Omega^d)$.
In other words, the cohomology groups $H^*(M,\mathbb{R})$ are isomorphic to the space of solutions of the equation $\Delta \omega = 0$.
Hodge theory is the application of elliptic PDE theory to this problem. It proves that there is a unique harmonic form in each cohomology class, and that the space of harmonic forms is finite dimensional. To follow the intuitive cohomology beats before, imagine we cut up the manifold into open balls. There are infinitely many harmonic forms on open unit balls. Harmonic forms are determined by their taylor series at any point. If you set the conditions at the origin, it is likely it will extrapolate to a harmonic form on the whole ball. However, once you start to glue balls together and have loops, you need the extrapolated harmonic forms to agree at the gluing. This cuts down the infinite dimensions of solutions s to a finite dimensional space.
This lets us approach Dolbeaut cohomology too. The cohomology groups $H^{p,q}(M)$ are defined to be the cohomology of the $\overline{\partial}$ operator applied to forms of type $p,q$: I.e, those which have $p$ holomorphic components, and $q$ antiholomorphic ones. Hodge theory once again identifies a unique form in each cohomology class, the kernel of the holomorphic laplacian $\Delta_{\overline{\partial}} = \overline{\partial}^* \overline{\partial} + \overline{\partial} \overline{\partial}^*$.
While the hodge theory perspective is enormously useful theoretically, it doesn’t much help us compute specific cohomology groups. The issue is the same as before: the condition that the laplacian of a form vanishes is a global condition, and it has global consequences on the structure of the form. It has to, if its ever going to measure something global like cohomology. It’s not at all clear how to glue together two harmonic forms. We need some way to simplify the differential equation on separate, understandable pieces…
[!thm]- Digression: Gluing formula I don’t mean to say it’s impossible to find gluing formula, just that its a lotttt of work. In gauge theory, which studies solutions to nonlinear elliptic equations much like the laplacian, there are plenty of ways to glue together two solutions from two halfs of the manifold. The argument generally starts by choosing a special Riemannian metric, which stretches out the region connecting the two halves into a long neck. (we can do this because cohomology is invariant under change of metric) Next we construct an approximate solution. On the two halves we have the starting solutions, and on the neck we find a solution which is invariant under the stretched out direction.
We sandwich these solutions on the three parts of the manifold one next to each-other, smoothing the hard edges to get a function. This is not a solution to the differential equation, but it gets pretty close, only failing near the borders of the neck. As we make the neck longer and longer, we can get this function arbitrarily close to a solution (in $L^2$). Once we check convergence, we can use some form of elliptic regularity to show the limit is actually a smooth solution. Thus, we have glued two solutions together.
The analysis behind this gets very hairy, espically because we rarely have the luxury of linear PDEs. Getting such a result is a whole, meaty paper.
A “What’s up” to Witten’s morse theory
To reconcile this, we use an inspired idea by Witten to combine morse theory with hodge theory. We rescale the laplacian $\Delta$ according to a morse function $f$, reducing the kernel of $\Delta$ to parts at each critical point of $f$? Namely, we consider a new operator \(\Delta_t = e^{t f} \Delta e^{-t f}\) $\Delta_t$ is a deformation of the usual laplacian. But, they have the same kernel, since the operator $e^{tf}$ is invertible! So, we can compute cohomology by solving $\Delta_t \omega = 0$ for any $t$. Why not take the limit $t\to \infty$? This makes the effect of $f$ get larger and larger, eventually overpowering any of the intrinsic geometry of the manifold. To see this explicitly, we compute the deformed laplacian \(\Delta_t = \Delta + t^2 \vert \nabla f \vert^2 + \Phi\) Here, $\Phi$ is a zeroth order operator acting on differential forms, which is uniformly bounded in $t$. (she doesn’t matter much). As $t$ goes to infinity, the term $\vert \nabla f \vert^2$ dominates everything. In particular, If a form $\omega$ has any hope of solving $\Delta_t \omega = 0$, then $\omega$ can’t be large where $\nabla f$ is large. This forces solutions to $\Delta_t \omega = 0$ to be concentrated very near the zeros of $\nabla f$ when $t$ is large.
To be more precise, the spectrum of $\Delta_t$ consists of some set of real numbers for all $t$. The number of zero eigenvalues is the same for all finite values of $t$, but the nonzero eigenvalues change continuously with $t$. As $t$ goes to infinity, some number of these approach zero. These are the asymptotically zero-energy states. Though we really want the states which are zero energy for all $t$, the asymptotically zero-energy states are much easier to calculate.
Intuitively, asymptotically zero-energy states are concentrated at their own critical points, so they should decouple from one another. With the gumption and carefreeness known only to a physicist, Witten contents himself with a “by perturbation theory”, and reduces to a local calculation. Near a critical point, $f$ is quadratic and the laplacian $\Delta_t$ reduces to the Schrodinger operator for a supersymmetric quantum harmonic oscillator. Something physicists understand very well. Turns out, the kernel of this local model is always one dimensional! Isn’t that convenient. The solution to $\Delta_t \omega=0$ in this local model is a differential form of degree equal to the index of the critical point.
To summarize: In the limit as $t \to \infty$, the asymptotic kernel of $\Delta_t$ has one dimension per critical point, and the degree of the form is the index of the critical point. We have done the first step of the cohomology intuition, splitting up the manifold into understandable parts. The dimension of the asymptotic kernel must certainly be greater than the dimension of the true kernel. So, just like in Morse theory, the dimension of the cohomology of degree $d$ is bounded above by the number of critical points with index $d$.
Now the gluing: which of these asymptotic zero-energy states are truly zero energy? How do these forms localized at the critical points of $f$ glue together? Witten approaches this with some standard asymptotic techniques from quantum physics, namely WKB analysis of quantum tunneling. And he finds, sure enough, that two states couple together to form a true zero-energy state if they share a gradient flow line between them. This reproduces the gluing formula from standard morse theory.
Witten’s approach to morse theory was and is very influential. Algebraically, the core idea is to deform a complex according to a parameter $t$, while preserving its cohomology. In the limit $t \to 0$, the complex vastly simplifies, making the cohomology much easier. This idea is evocative enough to inspire 50 other constructions. More specifically within geometry, it gives a way to localize all sorts of differential operators to fixed points.
More personally, I think it’s captivating. The idea is elegant enough to feel inevitable. The kind of math which, on a particularly wistful day, makes you sorry you didn’t come up with it first. But it’s only so natural because witten is a physicist. The rescaling argument is one manifestation of the ever-present idea of renormalization. This reduced the full spectral problem to a quadratic potential, also known as a harmonic oscillator. Well, thats basically the whole field of physics! It’s a perfect example of how the right perspective can inspire new math. This is why I love mathematical physics.