Instanton-phobia (part 1)

The goal of this sequence of posts is to talk about instantons and sketch the proof of a recent paper with Ali Daemi and Mike Miller Eismeier (https://arxiv.org/abs/2207.03631).  

I think most mere mortal mathematicians have math phobias. Here I don't mean a fear of giving talks or people finding out you don't know as much as you think you are tricking them into thinking you know, although each of those fears warrants its own blog post.  Instead I mean being afraid of mathematical concepts.  Earlier in my career I was really afraid of the knot Floer complex, $CFK^\infty$.  The pictures people drew never made sense to me, and I could not envision how anyone could wield this abstract nonsense in a way that would produce topology.  I didn't really make progress on this until I was a postdoc and was collaborating with Jen Hom and going through some old Ozsváth-Szabó papers with Allison Miller.  Sometimes a phobia is going from definitions to everyday use - right now, Adam Levine and Lisa Piccirillo are unknowingly helping me with my fear of performing handleslides in Kirby Calculus; my sincere apologies to them.  Those two are really unafraid to calculate and I am envious.  But sometimes a phobia can be just about grappling with the definitions; this is how I feel about Spin$^c$ structures, for instance.  They still make me uncomfortable, but I hope not forever.  

Right now, I want to address instantonphobia.  I think a lot of low-dimensional topologists are instanton-phobic.  *I get it.*  I still kind of am too.  You're in a talk and someone writes down the Chern-Simons functional $$CS(\alpha) = \frac{1}{8 \pi^2} \int_Y Tr(\alpha \wedge d\alpha + \frac{2}{3} \alpha \wedge \alpha \wedge \alpha)$$ and you might as well pull the fire alarm and escape.  I think part of it stems from objects we don't see while drawing pictures of knots - metrics, connections, Banach manifolds, etc.  Maybe you took a class on Riemannian geometry you never used, or maybe you haven't even seen a connection before, so why turn up the suffering now?  I think there is a misunderstanding about instanton gauge theory that you need to be a technical differential geometry wizard to use it.  The goal of these next few posts is to present some basic ideas with the goal of avoiding anything too technical, culminating in the proof from the work with Ali and Mike.  In this first post I will sketch the three-manifold side of the story.  For an alternate background on instanton theory and its backgrounds, I recommend reading this nice article in the Notices by Juanita Pinzón-Caicedo and Danny Ruberman. 

Three-manifolds: The key object in instanton gauge theory for a three-manifold $Y$ is a flat connection.  What's that?  Don't worry about the definition.  Think of it as a homomorphism from $\pi_1(Y)$ to $SU(2)$ (modulo conjugation).  If you don't like $SU(2)$, just pretend it's homomorphisms to some abstract mystery group; it won't really matter for these posts.  For homology spheres, we only want the non-trivial ones, and instanton Floer homology is morally generated by these.  Therefore, $I_*(S^3) = 0$ and if $I_*(Y) \neq 0$, then $\pi_1(Y)$ has a non-trivial $SU(2)$ representation.  For the Poincare homology sphere, there are two non-trivial flat connections.  We can also talk about three-manifolds with $H_1(Z) = \mathbb{Z}$.  In this case, the instanton Floer homology basically counts representations to $SO(3)$ that cannot be lifted to $SU(2)$.  Don't worry about this too much, because for this class of three-manifolds, Kronheimer-Mrowka show $I_*(Z) = 0$ if and only if $Z$ has a non-separating $S^2$, i.e. $Z = S^2 \times S^1 \# Y$.  (The key idea is that if $Z$ has no non-separating $S^2$, there is a taut foliation and so $Z$ can be embedded in a nice symplectic 4-manifold.  This forces the invariants of $Z$ to be non-trivial.)  

One great tool is that there is a surgery exact triangle for a knot in a homology sphere: $$\ldots \to I_*(Y) \to I_*(Y_0(K)) \to I_*(Y_1(K)) \to \ldots $$

Actually, now we already know enough to prove something deep: Property P.  This says that surgery on a non-trivial knot in $S^3$ has non-trivial fundamental group.  This problem was old (i.e. pre-Poincare conjecture).  Applying the cyclic surgery theorem of Culler-Gordon-Luecke-Shalen and possibly reversing orientations, we can restrict to the case of $+1$-surgery.  If $S^3_1(K)$ had trivial fundamental group, then there are no non-trivial representations to $SU(2)$, so $I_*(S^3_1(K)) = 0$.  Therefore, $I_*(S^3_0(K)) = 0$ by the exact triangle.  We said above that means that $S^3_0(K)$ has a non-separating 2-sphere.  Gabai proved using foliations that 0-surgery on a non-trivial knot never has a non-separating 2-sphere, so $K$ had better be trivial.  Pretty cool, right?  (Kronheimer-Mrowka have 3 proofs of Property P.  This argument is from: https://arxiv.org/abs/math/0311489.  I highly recommend everyone read this one as well: https://arxiv.org/abs/math/0312322.  My personal philosophy is that there's still a lot to be extracted from the ideas here.) 

Let me now mention something that makes instantons stand out over other invariants like Heegaard Floer homology.  Associated to each flat connection / representation is a number, really an element of $\mathbb{R}/ \mathbb{Z}$, called the Chern-Simons invariant.  The set of flat connections with these numbers is a *topological* invariant.  If the Poincare homology sphere is oriented as the boundary of the negative definite $E_8$ plumbing, then the two representations can be told apart by their Chern-Simons invariants: 1/120 and 49/120.  Reversing orientation reverses the sign of the Chern-Simons invariants, and so we see that the Poincare sphere cannot have an orientation-reversing diffeomorphism.  For funsies, let me explain where the 120 comes from.  

Given a connection $\alpha$ on $Y$, the Chern-Simons functional is $$CS(\alpha) = \frac{1}{8 \pi^2} \int_Y Tr(\alpha \wedge d\alpha + \frac{2}{3} \alpha \wedge \alpha \wedge \alpha).$$Gross, I know.  But, since it's some kind of integral over a manifold, you can see that the trivial connection ($\alpha = 0$) has trivial Chern-Simons invariant.  Also, if $\alpha$ is a flat connection on $Y$, and $\pi: \tilde{Y} \to Y$ is an $n$-sheeted cover, then $n \cdot CS(\alpha) = CS( \pi^* \alpha)$.  Let's consider the Poincare sphere.  Its universal cover is 120-sheeted and the lift of any flat connection $\alpha$ on the Poincare sphere is necessarily trivial.  Therefore, we see that $120 \cdot CS(\alpha) = 0 \pmod{\mathbb{Z}}$, hence the 120 in the denominators.  See, it's not so bad.  Cool open problem: are all Chern-Simons invariants rational?  (We just proved this holds for any three-manifold with finite fundamental group / spherical geometry.)

There's a lot of cool stuff you can do with the Chern-Simons invariant, even if you don't consider instanton Floer homology.  For example, this has been used in various papers of Matt Hedden, Paul Kirk, and Juanita Pinzón-Caicedo to study knot linear independence in the knot concordance group (see, e.g. https://arxiv.org/abs/1009.5361, https://arxiv.org/abs/1809.04186).  If you look at the second paper, there is not really any "technical gauge theory".  I also believe that there should be some cool applications to Dehn surgery theory coming out of the Chern-Simons invariants; I would like to see that.

In the next blog post I will discuss the four-manifold side of the story and how the Chern-Simons functional can be used to get some topological information.