# Categorification of Floer homology

Floer homology associates a vector space $HF^\ast(L_1,L_2)$ to any pair of Lagrangian submanifolds $L_1,L_2$ inside a symplectic manifold $X$. By a categorification of Floer homology, I mean a category $\mathcal H\mathcal F(L_1,L_2)$ whose Hochschild homology is isomorphic to $HF^\ast(L_1,L_2)$.

Example: Let $M$ be a closed symplectic manifold, and consider the diagonal $\Delta\subseteq M\times M^-$, which is Lagrangian. It self-Floer homology $HF^\bullet(\Delta,\Delta)$ is (conjectured to be) isomorphic to the Hochschild homology of the Fukaya category of $M$. Thus $\mathcal F(M)$ is a categorification $\mathcal H\mathcal F(\Delta,\Delta)$ of $HF^\ast(\Delta,\Delta)$. Note $HF^\ast(\Delta,\Delta)$ is a unital algebra, and expressing this unit as a Hochschild cycle of $\mathcal F(M)$ is of fundamental importance in the study of the Fukaya category.

Are there any other known circumstances under which a categorification of Floer homology exists?

Floer homology also has a product $HF^\ast(L_1,L_2)\otimes HF^\ast(L_2,L_3)\to HF^\ast(L_1,L_3)$, so it is natural to further ask that a categorification $\mathcal H\mathcal F$ have natural functor $\mathcal H\mathcal F(L_1,L_2)\times\mathcal H\mathcal F(L_2,L_3)\to\mathcal H\mathcal F(L_1,L_3)$ which upon applying Hochschild homology recovers the product on Floer homology.

What is a natural geometric description of a functor $\mathcal F(M)\times\mathcal F(M)\to\mathcal F(M)$ corresponding to the product on $HF^\ast(\Delta,\Delta)$?

• We now know that the map from the cohomology of the Thurston manifold to the Hochschild cohomology of the Fukaya category is not an isomorphism, because the mirror is not algebraic. Apr 30, 2017 at 23:37
• Regarding categorification: the most natural thing to look for is a mirror to the conjectures of Kapustin-Rozansky-Saulina on the B-side (see arXiv:0810.5415). Apr 30, 2017 at 23:42
• I also don't think it's possible to expect in general that the product on the Floer cohomology of a diagonal arises from a functor as you are asking: this is like asking for the (intersection) product on the homology of a manifold to arise from the structure of an H-space on the given manifold. In both cases, the degree of the product is given by the dimension, so it doesn't work out. The classical case of course offers a hint as to what one can hope for. Apr 30, 2017 at 23:46
• To elaborate on Mohammed's comment: an optimistic version of your question would be to seek a 3d lift of the A-model, just as Rozansky-Witten theory is a 3d lift of the B-model (ie recovers the B-model on compactifying on $S^1$, i.e. taking Hochschild homology of Hom categories). The latter only exists when the target is holomorphic symplectic. Indeed to get a 3d SUSY QFT giving a TFT the target needs to be hyperkahler, so a physicist would likely insist your $M$ be holomorphic symplectic. May 1, 2017 at 16:27
• There is a "3d A-model" studied in arxiv.org/abs/1002.4241 but it recovers the A-model (of a cotangent bundle) by reduction on an interval with suitable boundary conditions, not on a circle, so doesn't give what you want. Teleman and Dimofte have discussed informally the existence of a 3d A-model with properties like you want in the holomorphic symplectic setting, and loosely "3d mirror" to Rozansky-Witten theory, but nothing is written beyond Teleman's seminal work, cf his ICM, in the case M=pt/G.. May 1, 2017 at 16:30

I am sort of an armchair mathematician these days, but I suspect the answer to your first question is "no". At least in the hep-th literature categorification of Khovanov homology I have seen many instances of, Floer homology in general I know much less about. There are just so many:

• Floer homology
• Knot Floer homology
• Bordered Floer homology
• Instanton Floer homology
• Monopole Floer homology
• Heegard-Floer homolo
• etc.

So the theory specializes in many different ways, and they all have really great properties. However, there are simply too many and I have lost track.

On the physics side, I know the way to understand all of these is Chern-Simons Theory and a small amount of google-fu returns

I am not an expert in 3-manifolds, so I can't say whether they fit into your question or not. I know a lot of the mathematical work is all about showing these functors map nicely onto the properties of 3-manifolds so that these define invariants.

Here I'm just being a librarian and not really opening up the papers in any way. I can read through Audin's Morse Theory and Floer Homology and notice complexes jump around as you change the choice of height function. There's a nice course on YouTube by Denis Auroux.

• I don't understand what the point of listing all these Floer homologies is. Knot Floer homology is not a "Floer homology" as the OP defines it (it comes from a filtration induced by a knot on a Floer homology). But ultimately all of the rest of these Floer homologies do (or are expected to) come from a construction as the OP defines. Work of Manolescu-Woodward and Horton on symplectic instanton homology and the fact that Heegaard-Floer homology is "symplectic monopole Floer homology", together with the fact that bordered Floer homology is a computational scheme for Heegaard-Floer homology... May 1, 2017 at 4:11
• ...(albeit one that comes from an interesting extension of HF to a 2+1+1-dim'l "TQFT"), means that all of the examples you've given are in fact examples of Lagrangian Floer homology. I also don't understand the relevance of the Chern-Simons shoutout - sure, some of the examples of Floer homologies you gave use the CS functional, but the OP's question is entirely contained within Lagrangian Floer homology (where you don't need a CS functional to define anything). May 1, 2017 at 4:16
• @dvitek no idea May 1, 2017 at 6:51