D-brane charges and K-homology

It is argued that D-brane charge takes values in K-homology. For smooth manifolds with spin structure, this could explain why the phase factor Ω(x) calculated with a D-brane state x in IIB theory appears in Diaconescu, Moore and Witten’s computation of the partition function of IIA string theory. Diaconescu, Moore and Witten [1] recently compared the partition functions of IIA string theory and M-theory. They found agreement, amusingly interpreted as a derivation of K-theory from M-theory. Many subtle mathematical effects have to conspire for this agreement, so it seems important to explore further the building blocks of their results. K-theory will enter into the discussion in a variety of ways so here is a summary of K-theory in string theory. Witten [2] argued that D-brane charge should take values in K-theory, following earlier work by Cheung and Yin [5] and Moore and Minasian [4], and providing a mathematical setting for the results of Sen [6]. To be precise, the groupsK(X) are associated with D-branes in IIB(A) string theory on the spacetime X, respectively. In later work, Moore and Witten [3] argued that Ramond-Ramond fields should take values in K-theory as well, with K(X) classifying these fields in IIB(A) string theory. An important point in [1] is the computation of a phase factor Ω(x) associated with an element x which is a Ramond-Ramond field in IIA string theory on a manifold X. Somewhat surprisingly, this phase factor is computed from a D-brane state in IIB string theory, also associated with x by means of the above arguments. The perplexing appearance of IIB string theory in a computation that a priori involves only IIA theory is pointed out in footnote 14 in [1]. Regressing a little bit to Polchinski’s basic operational definition of D-branes [7], recall that string theory in the presence of a D-brane is näıvely defined by specifying a submanifold M of X and including open strings which have both ends on M. Suppose we act with a diffeomorphism