Quantization of the Hitchin system from loop operator expectation values

Abstract

The subject of this thesis is exact results on N = 2 supersymmetric quantum field theories. We restrict our attention to a certain class of quantum field theories in the literature referred to as class S. For these theories, we study partition functions on a deformed four-sphere in the presence of extra defect observables called loop and surface operators. In a degeneration limit of the four-sphere, we find that these partition functions are related to the eigenfunctions of the conserved quantities in certain quantum integrable models. We establish a direct correspondence between parameters classifying the eigenfunctions in the relevant integrable models and the charge labels of the loop operators. Furthermore, the problem to classify the eigenvalues is mapped to the mathematical problem of classifying what are called projective structures with real monodromy. We exhibit various points of contact between the mathematical description of these projective structures and known as well as new exact results on class S theories. Apart from providing new non-perturbative results on the theories of class S, one may obtain further support of the S-duality conjectures from the results presented here.