D-branes, K-theory, and Non-representable Cycles

The fundamental objects of study are D-branes, which are characterized by a
cycle and vector bundle over the cycle. We discuss a kind of bordism that
defines equivalence classes of D-branes and we argue that equivalences classes
correspond to twisted K-theory classes of the total space. As applications,
one is led to a twisted generalization of the notion of representable cycles
and a generalization of mirror symmetry called T-duality, which is an
isomorphism of the twisted K-theory.