Topological quantum D-branes and wild embeddings from exotic smooth R^4

TL;DR

This paper explores the connection between exotic smooth structures on R^4, wild embeddings, and topological quantum D-branes, proposing a new framework linking topology, string theory, and noncommutative geometry.

Contribution

It introduces a novel approach to relate wild embeddings and exotic smoothness of R^4 to quantum D-branes within string theory, using C*-algebras and topological actions.

Findings

Wild embeddings correspond to Dp-branes in string theory.

Constructed an action for topological quantum D-branes that reduces to the Born-Infeld action in the classical limit.

Linked exotic smooth structures to non-trivial NS-NS H-fluxes and D-brane charges.

Abstract

This is the next step of uncovering the relation between string theory and exotic smooth R^4. Exotic smoothness of R^4 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 Dp-branes as well to KK theory. These branes emerge when there are non-trivial NS-NS H-fluxes where the topological classes are determined by wild embeddings S^2 -> S^3. Then wild embeddings of higher dimensional $p$-complexes into S^n correspond to Dp-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.