Berkovits and Vafa have a paper that I choose to be the most interesting paper on the arXiv today. They may have made steps towards a full proof of the perturbative part of the most famous example of the AdS/CFT correspondence, namely between the N=4 gauge theory on one side and the AdS5 x S5 background of the type IIB string on the other side.
Some aspects of their construction are well-known to me but others are new. Among the aspects we have realized for some time, we can mention the fact that the relevant worldsheet description of the type IIB string is similar to the pure spinor language of Berkovits. It is an A-model which is a quotient of the U(2,2/4) supergroup and its maximal bosonic subgroup. Another aspect that I have believed for a few years is that the proof is based on a reduction of the two-dimensinoal worldsheet to Feynman diagrams, by "erasing" most of the information inside disk-shape regions included in the Feynman diagram.
What seems new, and what I am not able to fully check at this moment, is their statement that the proof could be analogous to the Ooguri-Vafa proof of a similar but topological Gopakumar-Vafa duality. The key feature that Berkovits and Vafa claim to be shared is that the worldsheet theory has a new Coulomb branch in this case much like it had in the topological case. The pieces of the worldsheet that are found in this Coulomb branch are interpreted as the faces of the Feynman diagrams while the boundaries between them are its propagators and vertices. As the 't Hooft coupling goes to zero, the regions found in the Higgs branch shrink and become the propagators and vertices.
Clearly, Cumrun Vafa is the first person who would be expected to suggest such a picture - in fact, I have heard such hints from him some time ago - and people who are less topological than he is, which surely includes your humble correspondent but most likely also all other humans on this planet, with a possible exception of Edward Witten and hypothetically also Marcos Mariňo (and with apologies to Aganagič, Gopakumar, Ooguri, Saulina, and others), face greater hurdles in trying to follow the details here.
At any rate, this Coulomb branch picture is different from what I wanted to do with the faces of the Feynman diagram most recently. I wanted to "integrate them out" by a Euclidean version of the AdS2/CFT1 correspondence. The faces would be interpreted as the Euclidean AdS2 whose geometry is a Poincaré disk but the partition sum over these disks as a function of the boundary conditions could be calculated by AdS/CFT methods, leading to a CFT1 theory that would match the action needed in the Schwinger parameterization of the N=4 gauge propagators. The vertices would then require a special treatment as additional singular places.
Now, is such a proof important? I surely don't think that we need a proof in order to be more certain about the correspondence because the correspondence is obviously true and only crackpots - such as those whose shoes are connected by a rope - doubt it at this moment. But I do think that such a proof would allow us to see the degrees of freedom in different descriptions of string theory from a more unified perspective. It would help us to see a little bit more why the degrees of freedom in different formulations of string theory look so different and how they can actually be directly transformed to each other.
And that would be certainly worth a memo.