KO-Homology and Type I String Theory

TL;DR

This paper develops a geometric and analytic KO-homology framework to classify D-branes and Ramond-Ramond fields in Type I string theory, linking topological invariants with physical charge assignments and anomaly cancellation.

Contribution

It introduces a new geometric-analytic KO-homology isomorphism and applies it to classify D-branes and RR fields, incorporating torsion charges and anomaly considerations.

Findings

Established an explicit isomorphism between geometric and analytic KO-homology.

Derived cohomological index formulas for RR charges.

Connected KO-homology invariants with physical D-brane charges and anomaly cancellation.

Abstract

We study the classification of D-branes and Ramond-Ramond fields in Type I string theory by developing a geometric description of KO-homology. We define an analytic version of KO-homology using KK-theory of real C*-algebras, and construct explicitly the isomorphism between geometric and analytic KO-homology. The construction involves recasting the Cl(n)-index theorem and a certain geometric invariant into a homological framework which is used, along with a definition of the real Chern character in KO-homology, to derive cohomological index formulas. We show that this invariant also naturally assigns torsion charges to non-BPS states in Type I string theory, in the construction of classes of D-branes in terms of topological KO-cycles. The formalism naturally captures the coupling of Ramond-Ramond fields to background D-branes which cancel global anomalies in the string theory path integral. We show that this is related to a physical interpretation of bivariant KK-theory in terms of decay processes on spacetime-filling branes. We also provide a construction of the holonomies of Ramond-Ramond fields in Type II string theory in terms of topological K-chains.