We review equivariant localization and through the Feynman formalism of quantum mechanics motivate its role as a tool for calculating partition functions. We also consider a specific supersymmetric theory of one boson and two fermions and conclude that by applying localization to its partition function we may arrive at a known result that has previously been derived using different approaches. This paper follows a similar article by Levent Akant.

[pt] Esta dissertação tem como objetivo exibir uma fórmula para cálcular o anel de cohomologia de um fibrado flag homogêneo de um grupo de Lie G compacto e conexo. Para concluir o resultado é usado a cohomologia equivariante, em particular, sua abordagem mais algébrica. Isto implica introduzir G- módulos e sua teoria equivariante, o que passa também por introduzir a álgebra de Weil, o modelo de Cartan e o homomorfismo característico. A demonstração do resultado também está fortemente baseada nas propriedades algébricas dos toros maximais de G. / [en] The purpose of this dissertation is to present a formula for calculating the cohomology ring of a homogeneous flag bundles of a compact and connected Lie G group. To conclude the result, the equivalent cohomology is used, in particular, its more algebraic approach. This implies introducing G modules and their equivalent theory, which also involves introducing Weil algebra, Cartans model and characteristic homomorphism. The income statement is also strongly based on the algebraic properties of the maximal torus of G.

In this thesis, we investigate supersymmetric quantum mechanics (SUSYQM) and its relation to index theorems and equivariant cohomology. We define some basic constructions on super vector spaces in order to set the language for the rest of the thesis. The path integral in quantum mechanics is reviewed together with some related calculational methods and we give a path integral expression for the Witten index. Thereafter, we discuss the structure of SUSYQM in general. One shows that the Witten index can be taken to be the difference in dimension of the bosonic and fermionic zero energy eigenspaces. In the subsequent section, we derive index theorems. The models investigated are the supersymmetric non-linear sigma models with one or two supercharges. The former produces the index theorem for the spin-complex and the latter the Chern-Gauss-Bonnet Theorem. We then generalise to the case when a group action (by a compact connected Lie group) is included and want to consider the orbit space as the underlying space, in which case equivariant cohomology is introduced. In particular, the Weil and Cartan models are investigated and SUSYQM Lagrangians are derived using the obtained differentials. The goal was to relate this to gauge quantum mechanics, which was unfortunately not successful. However, what was shown was that the Euler characteristics of a closed oriented manifold and its homotopy quotient by U(1)n coincide.