Monopole Floer Homology vs. Heegaard-Floer theory I have a (possibly very naive) question: what is the relation between Monopole Floer Homology and Heegaard-Floer theory? (both known and conjectured)


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*Is there some version of Atiyah-Floer conjecture that relates the two theories? 

*Is there some physical explanation to this phenomena?
 A: Ozsvath and Szabo constructed HF as a topological interpretation of SWF, and they noticed many links between the two. Roughly speaking, their Euler characteristics are the same, and the analog of the Atiyah-Floer conjecture is thus HF = SWF. As mentioned in the comments, there is a 5-part paper that develops such an equivalence (though still in peer-review phase?). The construction passes through a version of Embedded Contact Homology, a la Taubes' proof SWF = ECH.
Some further motivation/explanation for the equivalence is that Taubes has shown that SW theory on symplectic 4-manifolds is related to $J$-holomorphic curve theory (counts of monopoles are equal to counts of surfaces). During their graduate career, Hutchings and Lee developed a 3-dimensional version of this story: SW theory on 3-manifolds is equal to counts of gradient trajectories of $S^1$-valued Morse functions on 3-manifolds. It's the same spirit as Taubes' proof: Sequences of solutions to (perturbed) SW equations will limit to objects that are localized around flowlines/curves. These equivalences extend down one more dimension, where Taubes started it all: Vortices on surfaces (2-dimensional versions of SW theory) are in bijection with symmetric products of the surfaces. I think that based on this history, i.e. similar in spirit to Taubes' and Hutchings-Lee's construction, Tim Perutz has formulated a precise "Atiyah-Floer conjecture" for HF and SW on 3-manifolds (at least he has described this in recent seminar talks that I attended).
As for your last question in the comments, there is something similar: Bordered Monopole Floer Theory developed by Jon Bloom and John Baldwin (I think still in progress). Their theory uses a finite "generating" set of bordered (i.e. parametrized surface-boundary) handlebodies $\lbrace (H_\alpha,\partial H_\alpha\cong\Sigma)\rbrace$. There is a pairing $\mathcal{C}(Y_1\cup_\Sigma Y_2)\simeq_\text{quasi}\mathcal{C}(Y_1)\tilde{\otimes}_{\mathcal{A}(\Sigma)}\mathcal{C}(Y_2)$ using the $A_\infty$-tensor product, where $\mathcal{A}(\Sigma):=\bigoplus_{\alpha\beta}\widehat{CM}(H_\alpha\cup_\Sigma -H_\beta)$ and $\mathcal{C}(Y):=\bigoplus_\alpha\widehat{CM}(Y\cup_\Sigma -H_\alpha)$.
