APPLIED N=2 SUPERSPACE FORMALISM IN THREE DIMENSIONS

The three dimensional N = 6 Chern-Simons-Matter theory admits a superspace formulation. In particular, a similar formalism to the very well known four dimensional N = 1 superspace formalism exists. This is the three dimensional N = 2 formalism which can be thought as a complexification of the three dimensional N = 1 formalism. In this formalism, only the N = 2 supersymmetry is realized off-shell and more extended supersymmetries may be completely hidden or may be realized as flavor symmetries which do not commute with the supercharges. There are two main advantages of this formalism over the field theory approach which are immediately observed. Firstly, ultraviolet convergence of the diagrams is improved, and it is possible to derive many non-renormalization theorems for particular or general situations due to this improvement. Secondly, for any given calculation which can be compared to an equivalent field theory calculation, the amount of diagrams which are necessary to be calculated is dramatically reduced. One of the advantages of the component approach over the superspace approach is that, due to arguments alla Poggio-Quinn, since the component approach has only classically marginal couplings, it is known to be infrared safe. In the superspace approach, these arguments are no longer valid and either d = 4 super Yang-Mills theories formulated in N = 1 superspace or d = 3 Chern-Simons theories formulated in N = 2 superspace are plagued by infrared infinities. These infrared divergencies are an artifact of the formalism and are due to the gauge propagator structure. In the most studied case, which is N = 4 super Yang-Mills in N = 1 superspace, this infrared divergencies can be hidden with a safe gauge choice, but this is not the case for more general Yang-Mills theories. As we will show in this work, the Yang-Mills propagator emerges in quantum loop corrections of Chern-Simons theories and produces the same problem of infrared divergencies as in four dimensions. This whole thesis work is devoted to the application of N = 2 superspace techniques in general and in particular to the recent and highly interesting N = 6 supersymmetric Chern-Simons-matter theory. We detail the main characteristics of the formalism itself, we motivate Chern-Simons theories and we formulate its supersymmetric version. We review the application of the formalism to the calculation of super-Feynman diagrams, the general arguments of renormalization properties, and as a working example we study the two-loop renormalization of the gauge sector with and without matter. Then, we review the infrared flow of mass deformed N = 4 Yang-Mills theory to N = 3 Chern-Simons theory so as to derive the construction of ABJ(M) theory. Its full superspace and component formulations are given and compared and the relevance of this theory as a gauge dual of M-theory is discussed. After that, we study in deepness the problem of infrared divergencies of the formalism by calculating some of its Green functions. We provide a solution to this problem by proposing a non-canonical gauge fixing procedure and we show it at work. Finally we display the full power of the formalism by calculating the four-loop correction of the anomalous dimension of long operators which are relevant in the AdS/CFT context.