Ivancevic_Applied-Diff-Geom

746 Applied Differential Geometry: A Modern Introductiona D,0 , we must consider only those Sp(2r + 2, R) transformations in whichthe transformations of all fields are independent of a D,0 . These transformationsall leave a 0 invariant. There is no essential loss then in scaling the a’sso that a 0 = 1. Arrange the a D,i , a j with i, j = 1 . . . r as a column vectorv. The Sp(2r + 2, R) transformations that leave invariant a 0 = 1 act on vby v → Mv + c where M ∈ Sp(2r, R) and c is a constant. This is preciselythe duality group that we found in the global N = 2 theory.4.14.7 Structure of the Moduli Space4.14.7.1 Singularity at InfinityIt is actually quite easy to see explicitly the appearance of non–trivialmonodromies. In fact, asymptotic freedom implies a non–trivial monodromyat infinity. The renormalization group corrected classical formulaF one loop = iA 2 ln(A 2 /Λ 2 )/2π gives for large aa D = ∂F∂a ≈ 2iaπ ln(a/Λ) + ia π . (4.238)It follows that a D is not a single-valued function of a for large a. If werecall that the physical parameter is really u = 1 2 a2 (at least for large u anda), then the monodromy can be determined as follows. Under a circuit ofthe u plane at large u, one has ln u → ln u + 2πi, and hence ln a → ln a + πi.So the transformation is a D → −a D + 2a, a → −a. Thus, there is anon–trivial monodromy at infinity in the u plane,(( )) −1 2M ∞ = P T −2 =, (4.239)0 −1„ « 1 1where P is the element -1 of SL(2, Z), and as usual T = .0 1The factor of P in the monodromy exists already at the classical level.As we said above, a and -a are related by a gauge transformation (theWeyl subgroup of the SU(2) gauge group) and therefore we work on theu plane rather than its double cover, the a plane. In the anomaly freeZ 8 subgroup of the R symmetry group U(1) R , there is an operation thatacts on a by a → −a; when combined with a Weyl transformation, this isthe unbroken symmetry that we call P . Up to a gauge transformation itacts on the bosons by φ → −φ, so it reverses the sign of the low energyelectromagnetic field which in terms of SU(2) variables is proportional toTr (φF). Hence it reverses the signs of all electric and magnetic charges and