# How is Chern-Simons theory related to Floer homology?

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?

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.

• It is still unclear to me what you're asking... If you deal with instantons in some setup, you'll consider the CS functional, from which Floer theory could be applied. You can create vague relations (which probably can become precise), such as: instantons --> Seiberg-Witten equations --> monopole Floer --> Heegaard-Floer. Oct 11, 2018 at 16:54
• @ChrisGerig That is what I called a "superficial" relation in this question. I want to know deeper connection between them. Oct 11, 2018 at 17:33
• Floer homology is a tool, and will be applied when given the chance. CS theory pops up in multiple places, and Floer homology has a chance to be applied. I believe there is no deeper relation. Oct 11, 2018 at 17:45
• @ChrisGerig No, sorry but that's not my question. Maybe I should explain what I mean by "CS theory". Chern-Simons theory is the study of Chern-Simons TQFT obtained by quantizing the Chern-Simons Lagrangian. Jones polynomial, WRT invariant, and their various other refinement and categorification are considered parts of "CS theory". Floer homologies and instanton invariants are, at least traditionally, not considered CS theory. Oct 11, 2018 at 19:42
• @ChrisGerig In other words, I am asking how Chern-Simons theory, an instance of Schwarz type TQFT, is related to (various forms of) Floer homology, an instance of Witten type TQFT. Oct 11, 2018 at 19:44

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.