Ivancevic_Applied-Diff-Geom

750 Applied Differential Geometry: A Modern Introductionthe massless particle that produces a monodromy M has qM = q. Forinstance, monodromy M 1 arises from a massless monopole of charge vectorq 1 = (1, 0), and using the known form of M 1 , one has q 1 M 1 = q 1 . Dualitysymmetry implies that this must be so not just for the particular monodromyM 1 but for any monodromy coming from a massless particle. Uponsetting q −1 = (1, −1), we get q −1 M −1 = q −1 , and hence the monodromyM −1 arises from vanishing mass of a dyon of charges (1, −1).It seems that we are seeing massless particles of charges (1, 0) or (1, −1).However, there is in fact a complete democracy among dyons. The BPSsaturateddyons that exist semiclassically have charges (1, n) (or (−1, −n))for arbitrary integer n. The monodromy at infinity brings about a shift(1, n) → (1, n − 2). If one carries out this shift ntimes before proceeding tothe singularity at u = 1 or u = −1, the massless particles producing thosesingularities would have charges (1, −2n) and (1, −1 − 2n), respectively.This amounts to conjugating the representation of the fundamental groupby M n ∞ .The particular matrix A in (4.242) obeys A 2 = −1, which is equivalentto the identity as an automorphism of SL(2, Z). Conjugation byAimplements the underlying Z 2 symmetry of the quantum moduli space,which according to our assumptions, exchanges the two singularities. TheZ 2 maps M 1 → M 1 ′ = M −1 , M −1 → M −1 ′ = M 1 and M ∞ → M ∞ ′ =M 1M ′ −1 ′ = M −1 M 1 . Note that M ∞ ′ is not just obtained from M ∞ by conjugation,but the relation M ∞ = M 1 M −1 is preserved. The reason for thatis that (as in any situation in which one is considering a representation ofthe fundamental group of a manifold in a non–Abelian group), the definitionof the monodromies requires a choice of base point. The operationu → −u acts on the base point, and this has to be taken into account indetermining how M ∞ transforms under Z 2 .One can go farther and show that if one assumes the existence of a Z 2symmetry between M 1 and M −1 , then they must be conjugate to T 2 , andnot some other power of T . In our derivation of the monodromy (4.241),the 2 came from something entirely independent of the assumption of a Z 2symmetry, namely, from the charges and multiplicities of the monopolesthat exist semiclassically.4.14.7.5 Monopole Condensation and ConfinementWe recall that the underlying N = 2 chiral multiplet A decomposes underN = 1 supersymmetry as a vector multiplet W α and a chiral multiplet