7 research outputs found
Geometric K-Homology of Flat D-Branes
We use the Baum-Douglas construction of K-homology to explicitly describe
various aspects of D-branes in Type II superstring theory in the absence of
background supergravity form fields. We rigorously derive various stability
criteria for states of D-branes and show how standard bound state constructions
are naturally realized directly in terms of topological K-cycles. We formulate
the mechanism of flux stabilization in terms of the K-homology of non-trivial
fibre bundles. Along the way we derive a number of new mathematical results in
topological K-homology of independent interest.Comment: 45 pages; v2: References added; v3: Some substantial revision and
corrections, main results unchanged but presentation improved, references
added; to be published in Communications in Mathematical Physic
A survey of the homotopy properties of inclusion of certain types of configuration spaces into the Cartesian product
International audienceLet be a topological space. In this survey we consider several types of configuration spaces, namely the classical (usual) configuration spaces and , the orbit configuration spaces and with respect to a free action of a group on , and the graph configuration spaces and , where is a graph and is a suitable subgroup of the symmetric group . The ordered configuration spaces , , are all subsets of the -fold Cartesian product of with itself, and satisfy . If denotes one of these configuration spaces, we analyse the difference between and from a topological and homotopical point of view. The principal results known in the literature concern the usual configuration spaces. We are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusion , the homotopy type of the homotopy fibre of the map via certain constructions on various spaces that depend on , and the long exact sequence in homotopy of the fibration involving and arising from the inclusion . In this respect, if is either a surface without boundary, in particular if is the -sphere or the real projective plane, or a space whose universal covering is contractible, or an orbit space of the -dimensional sphere by a free action of a Lie Group , we present some recent results obtained in [23,24] for the first case, and in [18] for the second and third cases. We briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest. In order to motivate various questions, for the remaining types of configuration spaces, we describe and prove a few of their basic properties. We finish the paper with a list of open questions and problems