This is a very partial list of a few topics that appeared in very recent papers.

  • One important quantity in any QFT is the set of correlation functions between a operators. This is in general very hard to compute exactly. In some cases, a QFT can be deformed by a marginal operator that allows to interpolate between a weak coupling and a strong coupling regime, and the correlation functions change continuously during this interpolation. If it is possible to know exactly how they evolve, it becomes possible to know exactly the correlators at any coupling. In this respect supersymmetry lends a helping hand. Let us consider one of the simplest theories with $\mathcal{N}=2$ in $4d$, namely the $SU(2)$ superconformal QCD (this means that it is coupled to $N_f = 4$ hypermultiplets in the fundamental representation), where there is a single exactly marginal deformation, labeled by the complexified gauge coupling constant $\tau$. In this theory, the class of scalar superconformal chiral primary operators have a non-singular OPE (operator product expansion), which endows it with a ring structure called the chiral ring. Due to this structure, it is sufficient to know the 2- and 3-point functions as a function of $\tau$. This is what Baggio, Biarchos and Papadodimas have computed in 2014 (see this PRL). Note that this is the only example of exact computation of a non-trivial 3-point function in any $4d$ QFT (indeed, $\mathcal{N}=4$ is trivial in this respect)! The result is obtained using a combination of supersymmetric localization and a $4d$ analog of the $2d$ $tt^{\ast}$ equations of Cecotti and Vafa (1992, PRL again). In a paper that appeared yesterday, Baggio, Niarchos, Papadodimas, and Vos extended partially these results to $SU(N)$ SQCD in the large $N$ limit. The result is partial in two respects: first, the 3-point functions are restricted to a certain class of operators (for instance, the single-trace operators), and secondly, the results are obtained in the large and small 't Hooft coupling regimes as a perturbative expansion (that can be in principle extended to arbitrary order).
  • The Gribov problem is the object of a paper by Canfora, Hidalgo, and Pais, and it is a good reason to say a few word about this subtlety of non-Abelian gauge theories. Note that there is a surprisingly nice summary on Wikipedia, and you can also find a good review here. In one sentence, the problem is the following. Perturbation theory in non-abelian gauge theories (Feynman diagrams, etc) is tractable if there is a way to fix the gauge ambiguity, which can be done using the Fadeev-Popov procedure (FP), introducing the well-known ghosts. However in 1978 Gribov showed that the FP procedure does not work well non-perturbatively, since there exist gauge equivalent configurations (called the Gribov copies) that satisfy the gauge fixing condition, and this extends to a very large class of gauge-fixing procedure.
  • I signal quickly the surprising number theoretic problems that can appear when you consider a $3d$ Yang-Mills theory on a flat torus, reported by Chamizo, and Gonzalez-Arroyo (arxiv here).