Categorification of Chern-Weil theory and equivariant cohomology

Abstract

Esta tesis contempla la generalización de resultados de geomtría diferencial clásica en el contexto de los sistemas locales homotópicos. En particular, se realiza la construcción del homomorfismo de Chern-Weil y el teorema equivariante de de Rham en el contexto de las categorias diferenciales graduadas conformadas por los sistemas locales homotópicos. (Texto tomado de la fuente)

Let G be a compact connected Lie group acting on a smooth manifold M. We show that the DG categories Loc∞(BG) and Loc∞(MG) of ∞-local systems on the classifying space of G and the homotopy quotient of M, respectively, can be described infinitesimally as the categories InfLoc∞(g) of basic g-L∞ spaces and InfLoc∞(g,M) of g graded G-equivariant vector bundles, respectively. Moreover, we show that, given a principal bundle π : P → X with structure group G and any connection θ on P, there are DG functors C Wθ : InfLoc∞(g) −→ Loc∞(X), and Cθ : InfLoc∞(g,M) −→ Loc∞((P× M)/G), that corresponds to the pullback functor by the classifying map of P. An A∞-natural isomorphism relates the functors associated with different connections. This construction categorizes the ChernWeil homomorphism, which is recovered by applying the functor C Wθ to the endomorphisms of the constant local system. Finally, we obtain a categorification of the equivariant de Rham theorem for infinity local systems, namely, the A∞-fuctor DR : InfLoc∞(g,M) → Loc∞(MG).