D-branes and $K$-homology

TL;DR

This thesis explores the relationship between topological K-homology and D-branes in string theory, proposing that K-cycles classify D-brane configurations and their charges, including those wrapping singular subspaces.

Contribution

It introduces a K-homology framework for classifying D-branes, linking mathematical K-cycles to physical D-brane configurations and charges, including singular cases.

Findings

K-homology elements correspond to D-brane configurations.

K-cycle classification includes D-branes wrapping singular subspaces.

Provides a new mathematical perspective on D-brane charges.

Abstract

In this thesis the close relationship between the topological $K$-homology group of the spacetime manifold $X$ of string theory and D-branes in string theory is examined. An element of the $K$-homology group is given by an equivalence class of $K$-cycles $[M,E,\phi]$, where $M$ is a closed spin$^c$ manifold, $E$ is a complex vector bundle over $M$, and $\phi: M\rightarrow X$ is a continuous map. It is proposed that a $K$-cycle $[M,E,\phi]$ represents a D-brane configuration wrapping the subspace $\phi(M)$. As a consequence, the $K$-homology element defined by $[M,E,\phi]$ represents a class of D-brane configurations that have the same physical charge. Furthermore, the $K$-cycle representation of D-branes resembles the modern way of characterizing fundamental strings, in which the strings are represented as two-dimensional surfaces with maps into the spacetime manifold. This classification of D-branes also suggests the possibility of physically interpreting D-branes wrapping singular subspaces of spacetime, enlarging the known types of singularities that string theory can cope with.