# Yang-Mills/topological string theory (M-theory) duality

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.