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D-branes and $K$-homology

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 $[M,E,ϕ]$, where $M$ is a closed spin$^c$ manifold, $E$ is a complex vector bundle over $M$, and $ϕ: M\rightarrow X$ is a continuous map. It is proposed that a $K$-cycle $[M,E,ϕ]$ represents a D-brane configuration wrapping the subspace $ϕ(M)$. As a consequence, the $K$-homology element defined by $[M,E,ϕ]$ 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.