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

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.