Ivancevic_Applied-Diff-Geom

734 Applied Differential Geometry: A Modern Introductionthe form τ = θ2π + i 4πg, and denote the effective couplings in the vacuum2parametrized by a as τ(a), then τ(a) = ∂2 F∂a. 24. The generalization to an arbitrary compact gauge group G of rankr is as follows. The potential is always given by (4.214), so the classicalvacua are labelled by a complex adjoint–valued matrix φ with [φ, φ † ] = 0.The unbroken gauge symmetry at the generic point on the moduli space isthe Cartan subalgebra and therefore the complex dimension of the modulispace is r. The low energy theory is described in terms of r Abelian chiralmultiplets A i , and the generalization of (4.215) is[∫14π Imd 4 θ ∂F(A)∂A i Ā i +∫d 2 θ 1 ∂ 2 ]F(A)2 ∂A i ∂A j W αW i αj . (4.217)Here i labels the generators in the Cartan subalgebra and locally F is anarbitrary holomorphic function of r complex variables.5. The SU(2) theory, studied on the flat direction with u ≠ 0, has inaddition to the massless chiral or vector multiplet A, additional chargedmassive vector multiplets. One can easily write a gauge invariant effectiveaction for the triplet of chiral multiplets A a , a = 1 . . . 3, which reduces atlow energies to (4.215) for the massless fields and incorporates the massiveones. Using the same function F as above, we set F( √ A · A) = H(A · A)and write[∫12π Imd 4 θH ′ A a ( e V ) ∫ab Āb +(H ′ δ ab + 2H ′′ A a A b) ]WαW a αb ,d 2 θ 1 2(4.218)where the SU(2)−invariant metric δ ab has been used to raise and lowerindices. (4.218) has N = 2 supersymmetry and manifest gauge invariance,and reduces at low energies to (4.218).6. The Lagrangian (4.218) is unchanged if we add to F terms linear inA. This has the effect of shifting ∂F/∂A by a constant.As already mentioned, classically the F function isThe one–loop contributions add up toF 0 = 1 2 τ clA 2 . (4.219)F one−loop = i 12π A2 ln A2Λ 2 , (4.220)where Λ is the dynamically generated scale. This logarithm is related to theone–loop beta function and also ensures the anomalous transformation laws