Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 743see this, let us see how duality works in Lagrangians of the sort introducedabove.We work in Minkowski space and consider first the purely bosonicterms involving only the gauge fields. We use conventions such thatF 2 µν = −( ∗ F) 2 µν and ∗ ( ∗ F) = −F where ∗ F denotes the dual of F. Therelevant terms are∫132π Imτ(a) · (F + i ∗ F) 2 = 116π Im ∫τ(a) · (F 2 + i ∗ FF). (4.233)Duality is carried out as follows. The constraint dF = 0 (which in theoriginal description follows from F = dA) is implemented by adding aLagrange multiplier vector field V D . Then F is treated as an independentfield and integrated over. The normalization is set as follows. The U(1) ⊂SU(2) is normalized such that all SO(3) fields have integer charges (mattermultiplets in the fundamental representation of SU(2) therefore have halfinteger charges). Then, a magnetic monopole corresponds to ɛ 0µνρ ∂ µ F νρ =8πδ (3) (x). For V D to couple to it with charge one, we add to (4.233)∫1V Dµ ɛ µνρσ ∂ ν F ρσ = 1 ∫∗ F D F = 1 ∫8π8π16π Re ( ∗ F D − iF D )(F + i ∗ F),where F Dµν = ∂ µ V Dν − ∂ ν V Dµ is the the field strength of V D . We can nowperform the Gaussian functional integral over F and find an equivalentLagrangian for V D ,(1 −132π Im τ)(F D + i ∗ F D ) 2 = 1 ( −116π Im τ)(F 2 D + i ∗ F D F D ).We now repeat these steps in N = 1 superspace. We treat W α in∫18π Im d 2 θτ(A)W 2as an independent chiral field. The superspace version of the Bianchi identitydF = 0 isIm(DW ) = 0,where D is the supercovariant derivative. It can be implemented by a realvector superfield V D Lagrange multiplier. We add to the action∫14π Imd 4 xd 4 θV D DW = 14π Re ∫d 4 xd 4 θiDV D W = − 14π Im ∫d 4 xd 2 θW D W.