We report on some recent analytical results on the behaviour of the gluon and ghost propagators in Euclidean SU(2) Yang-Mills theory quantized in the maximal Abelian gauge (MAG). This gauge is of particular interest for the dual superconductivity picture to explain color confinement. Two kinds of effects are taken into account: those arising from a treatment of Gribov copies in the MAG and those arising from a dynamical mass originating in a dimension two gluon condensate. The diagonal component of the gluon propagator displays the typical Gribov-type behaviour, while the off-diagonal component is of the Yukawa type due to the dynamical mass. These results are in qualitative agreement with available lattice data on the gluon propagators. The off-diagonal ghost propagator exhibits an infrared enhancement due to the Gribov restriction, while the diagonal one remains unaffected.

We investigate a dynamical mass generation mechanism for the off-diagonal gluons and ghosts in SU(N) Yang-Mills theories, quantized in the maximal Abelian gauge. Such a mass can be seen as evidence for the Abelian dominance in that gauge. It originates from the condensation of a mixed gluon-ghost operator of mass dimension two, which lowers the vacuum energy. We construct an effective potential for this operator by a combined use of the local composite operators technique with algebraic renormalization and we discuss the gauge parameter independence of the results. We also show that it is possible to connect the vacuum energy, due to the mass dimension two condensate discussed here, with the non-trivial vacuum energy originating from the condensate <FONT FACE=Symbol>á</FONT>Aµ²<FONT FACE=Symbol>ñ</FONT>, which has attracted much attention in the Landau gauge.

The effects of the Gribov copies on the gluon and ghost propagators are investigated in SU(2) Euclidean Yang-Mills theory quantized in the maximal Abelian gauge. By following Gribov's original approach, extended to the maximal Abelian gauge, we are able to show that the diagonal component of the gluon propagator displays the characteristic Gribov type behavior. The off-diagonal component is found to be of the Yukawa type, with a dynamical mass originating from the dimension two gluon condensate, which is also taken into account. Furthermore, the off-diagonal ghost propagator exhibits infrared enhancement. Finally, we make a comparison with available lattice data.

The existence of the vector supersymmetry is analysed within the context of the finite temperature Chern-Simons theory.

The existence of the vector supersymmetry for three dimensional cohomological field theories is discussed. The example of the three dimensional topological model of Baulieu-Grossman [1] is considered in detail.