2d Gauge Theories

Sammanfattning: The Yang-Mills gauge theories play prominent role in modern high energy physics and the direct non-perturbative calculations present the main challenge in the field.Two dimensional gauge theories provide a nice playground where it is possible to carry out exact computations and at the same time some non-trivial phenomena canbe studied. This thesis is devoted to review and summarize a core portion of the quantum version of this field, accompanying the reader through the technicalities that one can encounter during the first approach to the subject.The thesis starts with a brief introduction to the subject and a brief review of the literature, the first two chapters are then spent on the computation of the partition functions over the simplest surfaces, i.e. disk and cylinder, using different formalisms.In chapter 4 there is presented the generalization of the procedure in order to find the general partition function that can be applied over any orientable surface (not necessarily closed). Interesting observables in this theory are the so called Wilson loops, chapter 5 contains the definition and a presentation of the non-intersecting ones over closed surfaces. In the end some of these encountered formulae have been applied to U(1) and SU(N) groups, i.e. we derived some partition functions of Maxwell theory and quantum chromodynamics over non-trivial surfaces and we gave the tools to explore more complex cases. The appendices contain one a brief review of the mathematics, the other all the step by step computations performed during the proofs.