Frame bundle

About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.

Structure of Malicious Singularities

Abstract: In this paper, we investigate relativistic spacetimes, together with their singular boundaries (including the strongest singularities of the Big Bang type, called malicious singularities), as noncommutative spaces. Such a space is defined by a noncommutative algebra on the transformation groupoid Γ = Ē × G, where Ē is the total space of the frame bundle over spacetime with its singular boundary, and G is its structural group. We show that there exists the bijective correspondence between unitary representations of the groupoid Γ and the systems of imprimitivity of the group G. This allows us to apply the Mackey theorem to this case, and deduce from it some information concerning singular fibers of the groupoid Γ. At regular points the group representation, which is a part of the corresponding system of imprimitivity, does not have discrete components, whereas at the malicious singularity such a group representation can be a single representation (in particular, an irreducible one) or a direct sum of such representations. A subgroup K ⊂ G, from which—according to the Mackey theorem—the representation is induced to the whole of G, can be regarded as measuring the “richness” of the singularity structure. In this sense, the structure of malicious singularities is richer than those of milder ones.

Character-valued index theorems in supersymmetric string theories

Abstract: By using (1+1)-dimensional supersymmetric sigma models, the local expressions for various G*U(1) character-valued topological indices in closed supersymmetric string theories are constructed explicitly, where U(1) is the group of constant rotations along the closed string and G is the non-trivial automorphism group of the vector bundle or the tangent bundle over which the string sigma model is defined.