Ivancevic_Applied-Diff-Geom

526 Applied Differential Geometry: A Modern IntroductionK−theory.The Freed–Hopkins–Teleman Theorem [Freed (2001); Freed et. al.(2003)] states: There is an isomorphism of algebrasV k (G) ∼ = Kdim G+ζ(k)G(G),where the right hand side is the ζ(k)−twisted equivariant K−theory indegree dim(G). The group structure on the right–hand side is inducedfrom the multiplication map G × G → G.For an arbitrary compact Lie group G the level k is replaced by a classin H 4 (BG; Z) and the dual Coxeter number h(G) is pulled back from auniversal class in H 4 (BSO; Z) via the adjoint representation. The twistingclass is obtained from their sum by transgression.4.5.8 Application: K−Theory in String TheoryIn string theory, K−theory has been conjectured to classify the allowedRamond–Ramond field strengths; 10 and also the charges of stableD−branes.4.5.8.1 Classification of Ramond–Ramond FluxesIn the classical limit of type II string theory, which is type II supergravity,the Ramond–Ramond (RR) field strengths are differential forms. In thequantum theory the well–definedness of the partition functions of D−branesimplies that the RR–field strengths obey Dirac quantization conditionswhen space–time is compact, or when a spatial slice is compact and one considersonly the (magnetic) components of the field strength which lie alongthe spatial directions. This led twentieth century physicists to classify RRfield strengths using cohomology with integral coefficients.However, some authors have argued that the cohomology of space–timewith integral coefficients is too big. For example, in the presence of Neveu–Schwarz (NS) H−flux, or non–spin cycles, some RR–fluxes dictate the presenceof D−branes. In the former case this is a consequence of the super-10 Recall that Ramond–Ramond (RR) fields are differential–form fields in the 10Dspace–time of type II supergravity theories, which are the classical limits of type IIstring theory. The ranks of the fields depend on which type II theory is considered. AsJoe Polchinski argued in 1995, D−branes are the charged objects that act as sources forthese fields, according to the rules of p−form electrodynamics. It has been conjecturedthat quantum RR fields are not differential forms, but instead are classified by twistedK−theory.