Periodic Report Summary 1 - ADSCFTBNF (AdS/CFT beyond the N=4 SYM paradigm)
We showed that any N = 2 SCFT contains a closed to all loops purely gluonic SU(2,1|2) sector that is made only out of fields in the N = 2 vector multiplet. The AdS duals of the states in the purely gluonic SU(2,1|2) sector live classically in AdS5 ×S1. We gave a diagrammatic argument that in the planar limit the SU(2,1|2) Hamiltonian is identical to all loops to that of N = 4 SYM, up to a redefinition of the coupling constant λ --> f(λ). Thus, this sector is integrable and anomalous dimensions can be read off from the N = 4 ones up to this redefinition. What is more, we showed how f(λ) can be exactly obtained, by comparing the circular Wilson loop of the respective N = 2 theories to the N = 4 one using Pestun localization. We obtained the f(λ) for a large class of N = 2 SCFTs and we checked our results pertubatively (using Feynman diagrams) up to four loops as well as at the strong coupling (using AdS/CFT) for the leading contribution. More results and more checks are work in progress.
Following a more formal path, we used 5-brane junctions and topological strings to study the 5D TN SCFTs corresponding to the 5D uplift of the 4D strongly coupled gauge theories obtained by compactifying N M5 branes on a sphere with three full punctures. These theories have no Lagrangian description, and traditional techniques cannot be used for them. However, using 5-brane junctions we were able to derive their SW curves and Nekrasov partition functions. Through the q-deformed version AGTW, 5D superconformal indices can provide n-point functions of the q-deformed WN Toda theories. Taking the undeformed q--> 1 limit we obtained the partition functions of the 4D TN theories with non-Lagramgian description, as well as undeformed WN Toda three point correlators that they correspond to. We thus succeed in providing the complete (with no degenerations) three point functions of Toda with three primary insertions; a long standing open problem in Mathematical physics. We have also been able to check our result by comparing with the formula by Fateev and Litvinov where one of the three primary fields is semi-degenerate.