Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 747acts as -1 ∈ SL(2, Z). The P monodromy could be removed by (perhapsartificially) working on the a plane instead of the u plane.The main new point here is the factor of T −2 which arises at the quantumlevel. This factor of T −2 has a simple physical explanation in terms ofthe electric charge of a magnetic monopole. Magnetic monopoles labelledby (n m , n e ) have anomalous electric charge n e + θ eff2π n m. The appropriateeffective theta parameter is the low energy oneθ eff = 2π Re[τ(a)] = 2π Re( daDda)= 2π Re( ) daD /du.da/duFor large |a|, we have θ eff ≈ −4 arg(a), which can be understood from theanomaly in the U(1) R symmetry. The monodromy at infinity transformsthe row vector (n m , n e ) to (−n m , −n e − 2n m ), which implies that (a D , a)transforms to (−a D +2a, −a). The electric charge of the magnetic monopolecan in fact be seen in the formula for Z, which if we take a D from (4.238)and set a = a 0 e −iθ eff /4 (with a 0 > 0) isZ ≈ a 0 e −iθ eff /4{(n e + θ eff n m2π)+ in m( 2 ln a0 /Λ + 1π)}.The monodromy under θ eff → θ eff + 4π is easily seen from this formula totransform (n m , n e ) in the expected fashion. Obviously, this simple formuladepended on the semiclassical expression (4.238) for a D ; with the exactexpressions we presently propose, the results are much more complicated,in part because the effective theta angle is no longer simply the argumentof a.4.14.7.2 Singularities at Strong CouplingThe monodromy at infinity means that there must be an additional singularitysomewhere in the u−plane. If M ′ is the moduli space of vacua withall singularities deleted, then the monodromies must give a representationof the fundamental group of M ′ in SL(2, Z). Can this representation beAbelian? If the monodromies all commute with P T −2 , then a 2 is a goodglobal complex coordinate, and the metric is globally of the form (4.228)with a global harmonic function Im τ(a). As we have already noted, sucha metric could not be positive.The alternative is to assume a non–Abelian representation of the fundamentalgroup. This requires at least two more punctures of the u plane(in addition to infinity). Since there is a symmetry u ↔ −u acting on