A small variation of U ∈ SU (2) is an element U ( ε ) ∈ SU (2) , where ε is a small real parameter, such that U (0) = U . The quantity U ( ε ) U − 1 is closed to the identity in SU (2) , and can be parametrized as U ( ε ) U − 1 = e − iεα for a function α ∈ su (2) . Deriving along ε , one gets, with the notation δU = dU ( ε ) dε | ε $_{=0}$, ( δU ) U − 1 = − iα . This gives also U ( δU − $^{1}$) = iα . A small variation of σ is a R -valued function δσ = λ ∈ R . A small variation of φ is an element δφ ∈ C $^{2}$. A small variation of the fields B is a collection of functions β$_{µ}$ ∈ su (2) . In a more geometric language, β defines an element in Ω 1 ⊗ su (2) , the space of 1 -forms on space-time with values in the Lie algebra su (2) . This is the tangent space to the space A of su (2) - connection 1 -forms. We introduce the notation δB = β . A small variation of b takes place in the same space. Small variations of ψ$_{L}$ and ψ ′ L are functions in S ⊗ C $^{2}$. We will denote by δψ ′ L = ϕ such a variation. The Jacobian one has to compute is the determinant of the matrix Using equations (A1) and (A2), straightforward computations give This implies that the only non zero blocks in J are given by the following maps: J ( $_{U, η, B, ψ}$′ $_{L}$) =    δφ δU δφ δσ δφ δB δφ δψ ′ L δb δU δb δσ δb δB δb δψ ′ L δψ$_{L}$ δU δψ$_{L}$ δσ δψ$_{L}$ δB δψ$_{L}$ δψ ′ L    δφ = − ( $_{UδU}$− $^{1}$) Ue σ   0 1   + Ue $^{σ}$( δσ $_{)}$  0 1   δb = 2 i g ( d ( $_{UδU}$− $^{1}$) + g 2 i [ $_{b, UδU}$− 1 ]) + U ( δB ) $_{U}$− 1 δψ$_{L}$ = − ( $_{UδU}$− $^{1}$) $_{Uψ}$′ L + $_{Uδψ}$′ L δφ δU : su (2) → C 2 α ↦→ − iαφ δφ δσ : R → C 2 λ ↦→ λφ δb δU : su (2) → Ω 1 ⊗ su (2) α ↦→ 2 i g D$_{b}$α δb δB : Ω 1 ⊗ su (2) → Ω 1 ⊗ su (2) β ↦→ Ad$_{U}$ β δψ$_{L}$ δU : su (2) → S ⊗ C 2 α ↦→ − iαψ$_{L}$ δψ$_{L}$ δψ ′ L : S ⊗ C 2 → S ⊗ C 2 ϕ ↦→ Uϕ 14