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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 

<img src='content_image/328733.jpg'>

<img src='content_image/328732.jpg'>

This implies that the only non zero blocks in J are given by the following maps: 

<img src='content_image/328734.jpg'>

