Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 7239. It has a holomorphic structure10. It has a duality that connects the weak and strong coupling regimes12. The duality generalizes to an SL(2, Z) symmetry.4.14.1.3 Spontaneous Symmetry–BreakingFor SU(2) this concept is very simple. From the form of the Higgs potentialin (4.208) we see that there is a Higgs vacuum for φ = vσ where v isany complex number and σ is any fixed generator of SU(2). Furthermore,for v ≠ 0 this breaks the gauge-symmetry from SU(2) to U(1). For othergauge-groups G the corresponding statement is that v must lie in the Cartansubalgebra of G. On the other hand there is no spontaneous breakdownof supersymmetry. Thus the full breakdown isSU(2) → U(1) : N = 2 supersymmetry unbroken.Indeed it is the fact that the supersymmetry is unbroken that gives themodel its nice properties, since otherwise the classical properties would notbe preserved after quantization.After the spontaneous breakdown the restriction of the N = 1 form of theclassical action (4.207) to the massless U(1) fields takes the form∫) ∫ (A = Im d 4 xd 2 θ α d(ĀA 2¯θβ + τ o Im d 4 xd 2 θ α W α W α).Since the adjoint representation of U(1) is trivial this action is a free-fieldone. However, in the quantum theory this does not mean that the effectiveLagrangian is also free because, through the quantum fluctuations, themassive fields induce interaction term for the massless ones. The first greatvirtue of the SW model is that these interactions have a very specific form.In fact, they have shown that, due to the N = 2 supersymmetry the localpart of the effective Lagrangian can only be of the formA = 1 ∫) ∫)d 4 xd 2 θd 2¯θ(ĀAd −2ĀdA + Im d 4 xd 2 θ(τ(A))(W α W α ,(4.209)where A d = F ′ (A) and τ(A) = F ′′ (A),for some function F(A). Thus the effective Lagrangian is completely governedby the single function F(A). Note that (4.209) is very similar to theclassical action (4.207) which is the special case for which F(A) = 1 2 τ oA 2 .