Ivancevic_Applied-Diff-Geom

738 Applied Differential Geometry: A Modern Introductionwhere the ellipses represent instanton corrections and Λ Nf is the dynamicallygenerated scale of the theory with N f flavors. The metric isds 2 = Im(a ′ Dā′ )du dū and the dyon masses M 2 = 2|Z 2 | are expressed interms of Z = n e a + n m a D , where n m , n e are the magnetic and electriccharges, respectively.4.14.6 DualityNext, Seiberg and Witten performed SL(2, Z) duality transformation onthe low energy fields. Although they are non–local on the photon field A µ ,they act simply on (a D , a). Several new issues appeared when matter fieldswere present.First, consider the situation of one massive quark with mass m Nf andexamine what happens when a approaches m Nf / √ 2 where one of the elementaryquarks becomes massless. Loop diagrams in which this quarkpropagates make a logarithmic contribution to a D . The behavior neara = m Nf / √ 2 is thusa ≈ a 0 , a D ≈ c − i2π (a − a 0) ln(a − a 0 ),with a 0 = m Nf / √ 2 and c a constant. The monodromy 26 around a = a 0 is26 Recall that monodromy is the study of how geometrical objects behave as they ‘runaround’ a singularity. It is closely associated with covering maps and their degenerationinto ramification; the aspect giving rise to monodromy phenomena is that certain functionswe may wish to define fail to be single–valued as we ‘run around’ a path encircling asingularity. The failure of monodromy is best measured by defining a monodromy group:a group of transformations acting on the data that codes what does happen as we ‘runaround’.In the case of a covering map, we look at monodromy as a special case of a fibration,and use the homotopy lifting property to ‘follow’ paths on the base space X (we assumeit is path–connected, for simplicity) as they are lifted up into the cover C. If we followround a loop based at a point x ∈ X, which we lift to start at c above x, we end at somec ∗ again above x; it is quite possible that c ≠ c ∗ , and to code this, one considers theaction of the fundamental group π 1 (X, x) as a permutation group on the set of all c, asa monodromy group in this context.An analogous geometrical role is played by parallel transport. In a principal bundle Bover a smooth manifold M, a connection allows ‘horizontal’ movement from fibers abovea point m ∈ M to adjacent ones. The effect when applied to loops based at m is todefine a holonomy group of translations of the fiber at m; if the structure group of Bis G, it is a subgroup of G that measures the deviation of B from the product bundleM × G.