Ivancevic_Applied-Diff-Geom

1244 Applied Differential Geometry: A Modern Introductionstring theory [Connes et. al. (1997)]. This is an interesting example, eventhough it involves only the zero modes of the strings. One would muchlike to have a fully stringy version involving a noncommutative algebraconstructed using all of the modes of the string, not just the zero modes.At this stage, we do not know what is the right noncommutative algebrathat uses all of the modes. One concrete candidate is the ∗−algebra of openstring field theory, defined in terms of gluing strings together. If we call thisalgebra A st , it seems plausible that D−brane charge is naturally labelledby K(A st ). For a manifold of very large volume compared to the stringscale, we would conjecture that K(A st ) is the same as the ordinary K(X)of topological K−theory.K−Theory and RR–FieldsNaively speaking, an RR p−form field G p obeys a Dirac quantization lawaccording to which, for any p−cycle U in space–time,∫G p= integer. (6.361)2πUIf that were the right condition, then RR fields would be classified by cohomology.However, that is not the right answer, because the actual quantizationcondition on RR periods is much more subtle than (6.361). Thereare a variety of corrections to (6.361) that involve space–time curvature andthe gauge fields on the brane, as well as self-duality and global anomalies.The answer, for Type IIB superstrings, turns out to be that RR fieldsare classified by K 1 (X). For our purposes, K 1 (X) can be defined as thegroup of components of the group of continuous maps from X to U(N), forany sufficiently large N. This means that topological classes of RR fields onX are classified by a map U : X → U(N) for some large N. The relation ofG p to U is roughly G p ∼ Tr (U −1 dU) p , where we have ignored correctionsdue to space–time curvature and subtleties associated with self-duality ofRR fields [Fabinger and Horava (2000)].The physical meaning of U is not clear. For Type IIA, the analog isthat RR fields are classified by a U(N) gauge bundle (for some large N)with connection A and curvature F A , the relation being G p ∼ Tr F p/2A. Theanalog for M−theory involves E 8 gauge bundles with connection. Again,the physical meaning of the U(N) or E 8 gauge fields is not clear.The value of using K 1 to classify RR fields of Type IIB is that this givesa concise way to summarize the otherwise rather complicated quantization