Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 527gravity equation of motion which states that the product of a RR–flux withthe NS 3–form is a D−brane charge density. Thus the set of topologicallydistinct RR–field strengths that can exist in brane–free configurations isonly a subset of the cohomology with integral coefficients.This subset is still too big, because some of these classes are related bylarge gauge transformations. In QED there are large gauge transformationswhich add integral multiples of 2π to Wilson loops. 1111 Recall that in gauge theory, a Wilson loop (named after Ken Wilson) is a gauge–invariant observable obtained from the holonomy of the gauge connection around a givenloop. In the classical theory, the collection of all Wilson loops contains sufficient informationto reconstruct the gauge connection, up to gauge transformation. In quantum fieldtheory, the definition of Wilson loops observables as bona fide operators on Fock space(actually, Haag’s Theorem states that Fock space does not exist for interacting QFTs)is a mathematically delicate problem and requires regularization, usually by equippingeach loop with a framing. The action of Wilson loop operators has the interpretation ofcreating an elementary excitation of the quantum field which is localized on the loop. Inthis way, Faraday’s ”flux tubes” become elementary excitations of the quantum electromagneticfield.Wilson loops were introduced in the 1970s in an attempt at a non–perturbative formulationof quantum chromodynamics (QCD), or at least as a convenient collection of variablesfor dealing with the strongly–interacting regime of QCD. The problem of confinement,which Wilson loops were designed to solve, remains unsolved to this day. The factthat strongly–coupled quantum gauge field theories have elementary non–perturbativeexcitations which are loops motivated Alexander Polyakov to formulate the first stringtheories, which described the propagation of an elementary quantum loop in spacetime.Wilson loops played an important role in the formulation of loop quantum gravity, butthere they are superseded by spin networks, a certain generalization of Wilson loops.In particle physics and string theory, Wilson loops are often called Wilson lines, especiallyWilson loops around non–contractible loops of a compact manifold.A Wilson line W C is a quantity defined by a path–ordered exponential of a gauge fieldA µIW C = Tr P exp i A µdx µ .CHere, C is a contour in space, P is the path–ordering operator, and the trace Tr guaranteesthat the operator is invariant under gauge transformations. Note that the quantitybeing traced over is an element of the gauge Lie group and the trace is really the characterof this element with respect to an irreducible representation, which means thereare infinitely many traces, one for each irrep.Precisely because we’re looking at the trace, it doesn’t matter which point on the loopis chosen as the initial point. They all give the same value.Actually, if A is viewed as a connection over a principal G−bundle, the equation abovereally ought to be ‘read’ as the parallel transport of the identity around the loop whichwould give an element of the Lie group G.Note that a path–ordered exponential is a convenient shorthand notation commonin physics which conceals a fair number of mathematical operations. A mathematicianwould refer to the path–ordered exponential of the connection as ‘the holonomy of theconnection’ and characterize it by the parallel–transport differential equation that itsatisfies.