TOPOLOGICAL QUANTUM D-BRANES AND WILD EMBEDDINGS FROM EXOTIC SMOOTH ℝ4

Abstract

This is the next step of uncovering the relation between string theory and exotic smooth ℝ4. Exotic smoothness of ℝ4 is correlated with D6-brane charges in IIA string theory. We construct wild embeddings of spheres and relate them to a class of topological quantum D p-branes as well to KK theory. These branes emerge when there are nontrivial NS–NS H-fluxes on S3 where the topological classes are determined by wild embeddings S2→S3. Then wild embeddings of higher dimensional p-complexes into Sn correspond to D p-branes. These wild embeddings as constructed by using gropes are basic objects to understand exotic smoothness as well Casson handles. Next we build C⋆-algebras corresponding to the embeddings. Finally we consider topological quantum D-branes as those which emerge from wild embeddings in question. We construct an action for these quantum D-branes and show that the classical limit agrees with the Born–Infeld action such that flat branes = usual embeddings.