It is known that there is a duality between Chern-Simons theory on 3-fold $X$ and topological A-model on the cotangent bundle of this manifold, $T^*X$ (see, for example, the original paper by Witten, Chern-Simons Gauge Theory As A String Theory).

In Topological M-theory as Unification of Form Theories of Gravity Vafa and friends proposed a generalization of the aforementioned duality to the following set of theories:

  • Topological gauge theory on 4-fold $M$;
  • Topological A-model on twistor space of this manifold, $T(M)$;
  • Topological M-theory on certain bundle over $M$.

Moreover, it is stated there that there is a deformation of A-model which is equivalent to full Yang-Mills theory.

I couldn't find a mention of any of these equivalences in the literature. Could anybody recommend some?


It's a longstanding hope that the planar limit of $\mathcal{N}=4,$ $SU(N)$ Super Yang-Mills theory on a four manifold can be computed from the topological string B-model on $\mathbb{CP}^{3|4}$ (with N topological D5 branes wrapping $\mathbb{CP}^{3|4}$). Nevertheless the precise identification is still conjectural.

The hope was initiated by Witten in his famous Perturbative Gauge Theory As A String Theory In Twistor Space and further developed in N=2 strings and the twistorial Calabi-Yau . The basic observation is the fact that the planar scattering amplitudes for $\mathcal{N}=4,$ $SU(N)$, $d=4$ Super Yang-Mills localize into holomorphic curves once translated into $\mathbb{CP}^{3|4}$; generally speaking, given a Calabi-Yau threefold one can think on holomorphic curves as $D1$-instantons if there are some spacefilling $D5$-branes because $D(p-4)$ brane instantons appear as worldvolume instantons of $Dp$-branes. The introduction of N=2 strings and the twistorial Calabi-Yau is in itself a wonderful summary of the hints that point out the conjectural equivalence.

Another related interesting connection is the equivalence between the $\Omega$ deformation of the topological string A-model and a $N=(2,0)$, $U(1)$ twisted version of Yang-Mills theory over the same Calabi-Yau threfold; this fact is known as Crystal Melting/String duality in the physics literature. With some ideas arising from this connection, Daniel Jafferis was capable to compute the partition function for twisted $N=4$ Super Yang-Mills theory in a complex Kahler surface in his PhD thesis.

Finally, other hints have appeared in many subtle ways in the topological string literature. A very interesting one was given in the paper String Theory Origin of Bipartite SCFTs in relation to the bipartite graphs involved in the theory of The Amplitudihedron.


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