Ivancevic_Applied-Diff-Geom

528 Applied Differential Geometry: A Modern IntroductionThe p− form potentials in type II supergravity theories also enjoythese large gauge transformations, but due to the presence of Chern–Simons terms in the supergravity actions these large gauge transformationstransform not only the p−form potentials but also simultaneously the(p + 3)−form field strengths. Thus to get the space of inequivalent fieldstrengths from the forementioned subset of integral cohomology we mustquotient by these large gauge transformations.The Atiyah–Hirzebruch spectral sequence constructs twisted K−theory,with a twist given by the NS 3−form field strength, as a quotient of asubset of the cohomology with integral coefficients. In the classical limit,which corresponds to working with rational coefficients, this is preciselythe quotient of a subset described above in supergravity. The quantumcorrections come from torsion classes and contain mod 2 torsion correctionsdue to the Freed–Witten anomaly.Thus twisted K−theory classifies the subset of RR–field strengths thatcan exist in the absence of D−branes quotiented by large gauge transformations.4.5.8.2 Classification of D−BranesNow, in many applications one wishes to add sources for the RR fields.These sources are called D−branes. As in classical electromagnetism, onemay add sources by including a coupling CpJ 10−p of the p−form potentialto a (10 − p)−form current J 10−p in the Lagrangian (density). The usualconvention in the string theory literature appears to be to not write thisterm explicitly in the action.The current J 10−p modifies the equation of motion that comes fromthe variation of Cp. As is the case with magnetic monopoles in electromagnetism,this source also invaliditates the dual Bianchi identity as it isa point at which the dual field is not defined. In the modified equationof motion J p+2 appears on the left hand side of the equation of motioninstead of zero. For simplicity, we will also interchange p and 7 − p, thenthe equation of motion in the presence of a source isJ 9−p = d 2 C 7−p = dG 8−p = dF 8−p + H ∧ G p−1 .The (9−p)−form J 9−p is the Dp−brane current, which means that it isPoincaré dual to the world–volume of a (p + 1)−D extended object calledIn finite temperature QCD, the expectation value of the Wilson line distinguishesbetween the ‘confined phase’ and the ‘deconfined phase’ of the theory.