How is Chern-Simons theory related to Floer homology?

Question

Chern-Simons theory (say, with gauge group $G$) is the quantum theory of the Chern-Simons functional $$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$ while instanton Floer homology is roughly the Morse homology of the moduli space of $G$-connections on $M$ with the Chern-Simons functional $CS(A)$ as Morse function.

Other than the superficial fact that they both use the Chern-Simons functional $CS(A)$ in some way, is there any deeper connection between the Chern-Simons theory and the Floer homology?

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Added 10/11, just to clarify what I am asking :

There must be many different aspects of the connection between Chern-Simons theory and the Floer homology. For instance, let me list a few of what I've heard :

• (Kronheimer & Mrowka, 2010) For a link $K$, there is a spectral sequence whose $E_2$ page is the Khovanov homology of $K$ that abuts to the orbifold instanton homology $I^\natural (K)$.
• (Gukov, Putrov, Vafa) Both Heegaard Floer homology and Chern-Simons gauge theory arise naturally in the context of $6\text{d }\mathcal{N}=(0,2)$ SCFT.

I am interested in learning more about these connections, and also want to know other aspects of connection between CS theory and Floer theory; So I think my question is twofold : for one, I want a conceptual explanation of why the two theories should be related, and for two, I want to expand the above list of known facts revealing some deeper connection between the two theories.

Any comment is more than welcome.

Answer 1

I'm far from an expert, and I apologize if this is too basic / philosophical / vague.

In instanton Floer homology, the functional $CS(A)$ plays the role of the potential energy function for a $4$d field theory. In Chern-Simons theory, $CS(A)$ plays the role of the action for a $3$d field theory.

Let's consider the analogous situation in the original setting for Floer homology (as in Supersymmetry + Morse theory). We have a Riemannian manifold $(M,g)$ and a function $f: M \to \mathbb R$ , analogous to the Chern Simons functional. We have two options:

First, we can use $f$ as a potential. This corresponds to a $1$d field theory (or classical mechanical system) whose fields are $\gamma(t) \in {\rm Map}([a,b],M)$, and whose action is $S(\gamma) = \int |\dot \gamma(t)|^2 + f(\gamma(t))dt$. This theory is dependent on the metric $g$, but the "vacuum states" of the quantum mechanical system stay the same as we deform the metric-- the vector space of vacuum states is the analog of the Floer groups.

Second we can use $f$ as the action of a $0$d field theory. The fields are ${\rm Map}(*,M) = M$, and the action is just $f(m)$. Physically, this describes a static system. The integral of $\exp(if)$ is the analog of the Chern-Simons invariants.

These two theories are related in the sense that if we let kinetic energy term of the $1$d field theory $|\dot \gamma(t)|^2$ tend to zero (for instance by making the metric $g$ small), the "limit" should be related to the $0$d field theory. This is the limit where the potential energy is very large relative to the kinetic energy, and the dynamical system approaches a static system.
This is to say, the $0$d field theory is a "dimensional reduction" of the $1$d field theory.

Analogously, we expect the $3$d Chern Simons theory to be a dimensional reduction of the $4$d Donaldson/Floer theory. Dimensional reduction is closely related to decategorification, for instance by taking Euler characteristics. The Kronheimer-Mrowka theorem you mention implies that Khovanov homology has the same Euler characteristic as instanton Floer homology. So they both categorify the Jones Polynomial / Chern simons invariant, as this (very) heuristic picture would suggest.