Instanton-phobia (part 3)

 

In this post, we describe how ASD connections on four-manifolds relate to Chern-Simons theory on three-manifolds.

Suppose that we have a smooth four-dimensional cobordism $W : Y_1 \to Y_2$ between homology three-spheres and consider the trivial $SU(2)$ bundle on $W$.  Just like for closed manifolds, we can count ASD connections on $W$.  The very rough idea is that an ASD connection restricts to a flat connection on the $Y_i$, and we can use it to relate the Chern-Simons invariants of $Y_1$ and $Y_2$.  (I lied a little, but I explain the issue in a technical point below.)  At the end of this post, we'll show that there is no simply-connected homology cobordism from the Poincare sphere to itself.  Get pumped!

Let's be a little more precise with our gauge theory now.  For a connection $A$ on $W$, we can measure its energy:   $$ \mathcal{E}(A) = \int_W F_A \wedge F_A. $$  If you don't like the formula, there are two relevant facts: if $A$ is ASD, then $\mathcal{E}(A) \geq 0$ and equality happens if and only if $A$ is flat.  You can get a ton of mileage out of this.  It turns out that if $A$ is an ASD connection, $$\mathcal{E}(A) = CS(A\mid_{Y_1}) - CS(A \mid_{Y_2}) \pmod{1}.$$  (Think Stokes' theorem.)  This is totally awesome.  Why?  If $A$ is flat on $W$, then $CS(A\mid_{Y_1}) = CS(A \mid_{Y_2})$.  If $\rho_1$ is an $SU(2)$ representation on $Y_1$ whose Chern-Simons we understand, if we can extend to $\rho$ on $W$, then $CS(\rho\mid_{Y_2}) = CS(\rho_1)$.  Auckly used these ideas in this paper to compute Chern-Simons invariants for all Seifert fibered spaces.  

Exercise: Chern-Simons extends to disconnected three-manifolds.  If $Y = Y_1 \# Y_2$, attach a 3-handle and deduce that $CS(\rho_1 \# \rho_2) = CS(\rho_1) + CS(\rho_2)$ for $\rho_i$ $SU(2)$ representations for $Y_i$.  

To get some further applications of the above ideas, we'll need to look at some non-flat ASD connections on cobordisms.  

Technical points, which you can probably ignore if you prefer.  

1) When we work with cobordisms, we usually want to attach cylindrical ends to $W$.  This means attaching an isometric copy of $Y_1 \times (-\infty, 0]$ and $Y_2 \times [1,\infty)$ to $W$.  ASD connections can have weird behavior at $\pm \infty$, but we will always look at instantons with finite energy, and these have the property that they converge to a specific flat connection at the ends.  I'll conflate the boundary and the ends to keep things a little simpler in notation.  For example, I'll talk about connections on $Y \times I$ and their restrictions to the boundary, when really I'll mean connections on $Y \times \mathbb{R}$ and the limiting connections on the ends.  I'll also point out that the differential in Floer homology is defined in terms of counting ASD connections on $Y \times \mathbb{R}$, which is actually the same as counting downward gradient flowlines of the Chern-Simons functional.

2) In order for the moduli spaces of ASD connections to be nice smooth manifolds, one often needs to work with a perturbation of the ASD equation.  This requires some work, but we will ignore this issue, since it doesn't interfere with understanding the main ideas.   

Given $\alpha_i$ on $Y_i$, let $\mathcal{M}(W, \alpha_1, \alpha_2)$ denote the moduli space of ASD connections on $W$ interpolating between $\alpha_1$ and $\alpha_2$.  Elements of $\mathcal{M}(W, \alpha_1, \alpha_2)$ can have different energies, but they all agree mod 1, and connected components will consist of connections with the same energies.  We might look at $\mathcal{M}_x(W, \alpha_1, \alpha_2)$ which consists of those connections with energy $x$.  While we can arrange for these to be smooth manifolds, the dimension formula is a little more complicated than for closed four-manifolds :(  

These moduli spaces are pretty tough to calculate in general, but for the trivial cobordism, $Y \times I$, there are some more known results.  Let's discuss this example when $Y$ is the Poincare sphere.  Recall that, oriented appropriately, there are three flat connections we care about: the trivial one, $\theta$, and then two irreducibles, $\alpha, \beta$ with Chern-Simons invariants $\frac{1}{120}$ and $\frac{49}{120}$ respectively.  It turns out that $\mathcal{M}_{\frac{1}{120}}(Y \times I,\alpha, \theta)$ is a copy of $\mathbb{R}$.  (Really, it's one solution, but you can translate by $\mathbb{R}$.)  Just by comparing Chern-Simons, we know that $\mathcal{M}_x(Y \times I,\beta,\theta) = \emptyset$ for any $x < \frac{49}{120}$. 

One thing that is an interesting feature of these moduli spaces is that they are in general *not* compact.  There are two ways that the moduli space can fail to be compact on a four-manifold with cylindrical ends: bubbling and leaking.  Bubbling does not occur if the energy is small, and so I won't discuss that here.  (It's similar to sphere-bubbling in pseudoholomorphic curves.)  Leaking means that a sequence of ASD connections in $\mathcal{M}_x(W, \alpha_1, \alpha_2)$ can limit to a "broken" ASD connection - a concatenation of ASD connections on $Y_1 \times \mathbb{R}$, $\mathcal{M}(W, \beta_1, \beta_2)$, and $Y_2 \times \mathbb{R}$.  (This is similar to broken trajectories in Morse theory.)  As a simple example, there might be some leaking on the $Y_2$-side, and a sequence converges to a once-broken trajectory, thought of as an element of $\mathcal{M}_y(W,\alpha_1, \beta) \times \mathcal{M}_z(Y \times I, \beta, \alpha_2)$.  If we have leaking, the energy that "leaks" must be positive; in our example, this means that $z > 0$.  Lastly, like in Morse theory, close to a broken ASD connection is an honest ASD connection.

Let's use what we've discussed to show that there is no simply-connected homology cobordism from the Poincare sphere to itself.  Let $Y = \Sigma(2,3,5)$, oriented as the boundary of a negative-definite four-manifold, and let $W: Y \to Y$ be a homology cobordism, i.e. the inclusions of the boundary components into $W$ induce isomorphisms on homology.  It turns out that $\mathcal{M}_{1/120}(W, \alpha, \theta)$ is a 1-manifold.  (Compare this to the similar result above on $Y \times I$, which sounds reasonable since $W$ and $Y \times I$ are topologically similar?)  That means that the boundary of the compactification has an even number of points.  This is the key insight.  

 Let's see how this compactifies.  There is one broken ASD connection that comes from combining the unique ASD connection on $Y \times I$ from $\alpha$ to $\theta$ with energy 1/120 together with the trivial flat connection on $W$.  Because there are an even number of broken ASD connections, let's go looking for more.  We cannot have broken ASD connections involving any other moduli spaces on $Y \times I$ from $\alpha$ to $\alpha, \beta$, or $\theta$; indeed, we have positive energy in this moduli space because of leaking, but we can check that any other ASD instanton would have energy more than 1/120.  (Here we are using that the Chern-Simons values on $Y$ are 0, 1/120, and 49/120.)  Therefore, we have to see a broken ASD connection which comes from combining something in $\mathcal{M}_0(W, \alpha, \omega)$ and something in $\mathcal{M}_{1/120}(Y \times I, \omega, \theta)$.  But an energy zero instanton on $W$ is a flat connection!! It can't be trivial, since it restricts to be non-trivial on the incoming copy of $Y$.  Therefore, we found a non-trivial $SU(2)$ representation for $\pi_1(W)$.  Hence, $W: Y \to Y$ can't be a simply-connected homology cobordism.  Another win for instantons!