Periodic Report Summary 1 - ERGTB (Exact Results in Gauge Theory and Beyond) Quantum field theory (QFT) plays a prominent role in our understanding of many phenomena in physics. In particle physics, QFT successfully combines special relativity with quantum mechanics and together with the gauge principle, constitutes the backbone of the standard model.The standard approach to quantum field theory relies on perturbation theory that is a powerful tool only when the system is weakly coupled. Indeed, for most of the models, exact results are not accessible, so that we cannot make any prediction for the strong coupling regime. This is certainly unsatisfactory from a theoretical point of view and also a limitation for phenomenological applications. For instance, this is the reason why the confinement of the strong interaction cannot be proved from first principles. It is therefore highly demanded to obtain a better understanding of the strongly coupled dynamics of quantum fields and as often happens in science, much progress can be made by attacking the problem using many different strategies.This research project has focused on the study and the development of few of the novel tools that have been recently introduced to investigate QFT at strong coupling. The novel tools include supersymmetric localization and several interconnections between gauge theory, 2D CFT, matrix models and string theory. In particular, the research directions pursued during the project, can be schematically summarized by the following points:A) Supersymmetric localization: This is a technique that enables to reduce the path integral that defines the partition function of a non-topological supersymmetric gauge theory to a finite dimensional matrix integral, much simpler to analyze. For instance, this result is extremely useful to the study non-perturbative dualities among gauge theories.B) The AGT correspondence, a connection between gauge theories and 2d CFT:For the case of N = 2 SU(N) gauge theory in four dimensions, it has been shown that partition functions and the VEV of certain observables, are equivalent to correlation functions in A_{N_1} Toda CFT in two dimensions. This relation rephrases some of the features of the four dimensional gauge theory in terms of two dimensional CFT. C) The AdS/CFT duality: This is a weak/strong duality between a string theory and a gauge theory. The strong coupling regime of the gauge theory is encoded by the dynamics of weakly coupled strings. The results obtained during the research project have been collected in the following publications:1) Exact results for vortex loop operators in 3d supersymmetric theories, Nadav Drukker, Takuya Okuda and Filippo PasseriniPublished in Journal of High Energy Physics 1407 137 (2014)This article focuses on three dimensional supersymmetric gauge theories, which are known to show many interesting non-perturbative dualities, like Seiberg-like dualities and mirror symmetry. In this article, using localization techniques we compute the expectation value of certain disorder line operators in three dimensional theories, dubbed vortex loop operators. They are defined by the path integral in the presence of prescribed singularities along the defect line and describe the coupling to the theory of an external, non-dynamical point-like vortex. Using these results, we are able to study how line operators behave under the action of dualities, providing significant insights for the strongly coupled dynamics of the three dimensional quantum fields. 2) 3d & 5d gauge theory partition functions as q-deformed CFT correlators, Fabrizio Nieri, Sara Pasquetti and Filippo Passerini,Published in Letters in Mathematical Physics 105 (2015) 1, 109-148.In this article, we show that partition functions for three dimensional and five dimensional supersymmetric gauge theories on compact manifolds are equivalent to certain q-deformed CFT correlators, that we have defined. The symmetry of the correlators is given by direct products of a q-deformation of the Virasoro algebra and the explicit expression of the correlators can be obtained applying the bootstrap approach. In particular, three-point functions are derived requiring invariance under crossing symmetry that follows from associativity of the operator algebra, and exploiting properties of degenerate representations of the q-deformed Virasoro algebra. This correspondence represents a novel tool to investigate the non-perturbative dynamics of three dimensional and five dimensional gauge theories. 3) 5D partition functions, q-Virasoro systems and integrable spin-chains Fabrizio Nieri, Sara Pasquetti, Filippo Passerini and Alessandro Torrielli Published in Journal of High Energy Physics 1412 (2014) 040 In this article, we analyze more in details the correspondence between 5d and 3d gauge theory partition functions and the q-CFT correlator that we introduced in our previous paper. In particular, we show that the partition function for 5-dimensonal N = 1 theories on S5 and S4xS1 can be written in terms of a set of fundamental 5d holomorphic blocks. We demonstrate that, when the 5d mass parameters are analytically continued to suitable values, the S5 partition functions degenerate to those for the 3-dimensional theory on S3. In a similar way, S4xS1 partition functions degenerate to the partition functions of 3-dimensional theory on S2xS1. We explain this mechanism via the correspondence between 5d partition functions and correlators with underlying q-Virasoro symmetry. From the q-Virasoro 3-point functions, we axiomatically derive a set of associated reflection coefficients, and show that they can be geometrically interpreted in terms of Harish-Chandra c-functions for quantum symmetric spaces. We link these particular c-functions to the types appearing in the Jost functions encoding the asymptotics of the scattering in integrable spin-chains, obtained taking different limits of the XYZ model to XXZ-type. These results introduce novel relations between q-Virasoro systems, 5d gauge theories and integrable models that deserve to be further analyzed in the future.
