Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 529a Dp−brane. The discrepency of one in the naming scheme is historicaland comes from the fact that one of the p + 1 directions spanned by theDp−brane is often time–like, leaving p spatial directions.The above Bianchi identity is interpreted to mean that the Dp−braneis, in analogy with magnetic monopoles in electromagnetism, magneticallycharged under the RR p−form C7−p. If instead one considers this Bianchiidentity to be a field equation for Cp + 1, then one says that the Dp−braneis electrically charged under the (p + 1)−form Cp + 1.The above equation of motion implies that there are two ways to derivethe Dp−brane charge from the ambient fluxes. First, one may integratedG 8−p over a surface, which will give the Dp−brane charge intersected bythat surface. The second method is related to the first by Stoke’s Theorem.One may integrate G 8−p over a cycle, this will yield the Dp−brane chargelinked by that cycle. The quantization of Dp−brane charge in the quantumtheory then implies the quantization of the field strengths G, but not of theimproved field strengths F .It has been conjectured that twisted K−theory classifies classifiesD−branes in noncompact space–times, intuitively in space–times in whichwe are not concerned about the flux sourced by the brane having nowhereto go. While the K−theory of a 10D space–time classifies D−branes assubsets of that space–time, if the space–time is the product of time anda fixed 9−manifold then K−theory also classifies the conserved D−branecharges on each 9D spatial slice. While we were required to forget aboutRR potentials to get the K−theory classification of RR field strengths,we are required to forget about RR field strengths to get the K−theoryclassification of D−branes.We will continue exposition on K−theory applications to string theoryin the last section of the book.4.6 Principal BundlesRecall that a principal bundle is a special case of a fibre bundle where thefibre is a group G. More specifically, G is usually a Lie group. A principalbundle is a total bundle space Y along with a surjective map π : Y → Xto a base manifold X. Any fibre π −1 (x) is a space isomorphic to G. Morespecifically, G acts freely and transitively without fixed point on the fibers,and this makes a fibre into a homogeneous space. It follows that the orbitsof the G−action are precisely the fibers of π : Y → X and the orbit space