14 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism The corresponding symplectic matrix (5.16) is rewritten as potential, The first-order correction term in θ , G $^{(1)}$, is determined by, which is obviously singular. Consequently, it has a zero- mode that, by construction, is given by equation (5.19). Now we start the second step of the method reformu- lating the model as a gauge invariant theory. At this stage, the correction terms as functions of θ , embraced by the arbitrary function G , given in Eq. (2.11), will be computed. It is achieved just imposing that no more constraints arise from the contraction of the zero-mode, given in Eq. (5.19), with the gradient of the symplectic ˜ f (0) =    0 − 1 0 0 0 q∂$_{y}$ 1 0 0 0 0 q 0 0 0 0 0 0 0 0 0 0 − 1 q 2 0 0 0 1 0 ∂$_{y}$ − q∂$_{x}$ − q 0 − q 2 − ∂$_{x}$ 0   $^{δ}$$^{(}$$^{x}$$^{−}$$^{y}$$^{) (5.21)}$ ν (0) α ∂ ˜ U (0) ∂ξ (0) α = 0 . (5.22) G $^{(1)}$( φ, π$_{φ}$, A$_{1}$, π$_{1}$, A$_{0}$, θ ) = − Ω$_{1}$ θ + q $^{2}$( a − 1) A ′ $_{1}$θ − q $^{2}$θπ$_{1}$ , (5.23) after an integration. Substituting this expression into the equation (5.13), the new Lagrangian is obtained as ˜ L (0) = π$_{φ}$ ˙ φ + π$_{1}$ ˙ A$_{1}$ + Ψ ˙ θ − ˜ U $^{(0)}$, (5.24) ˜ U (0) = 1 2 ( π 2 1 + π 2 φ + φ ′ $^{2}$) − A$_{0}$ [ π ′ 1 + 1 2 q $^{2}$( a − 1) A$_{0}$ + q $^{2}$A$_{1}$ + qπ$_{φ}$ + qφ $^{′}$] − A$_{1}$ [ − qπ$_{φ}$ − 1 2 q $^{2}$( a + 1) A$_{1}$ − qφ $^{′}$] − Ω$_{1}$ θ + q $^{2}$( a − 1) θ $^{′}$A$_{1}$ − q $^{2}$θπ$_{1}$ . (5.25) ν (0) α ∂ ˜ U (0) ∂ξ (0) α = − q $^{2}$( a − 1) θ + q $^{2}$( a − 1) θ ′′ + q $^{4}$θ − ∂ G (2) ∂θ = 0 , ⇒ G (2) = − 1 2 q $^{2}$( a − 1) θ ′ 2 + 1 2 q $^{4}$θ2 − 1 2 q $^{2}$( a − 1) θ $^{2}$. (5.27) ν (1) α ∂ ˜ U (0) ∂ξ (0) α = q $^{2}$( a − 1) θ ′′ − q $^{2}$( a − 1) θ + q $^{4}$θ, (5.26) ˜ L (0) = π$_{φ}$ ˙ φ + π$_{1}$ ˙ A$_{1}$ + Ψ ˙ θ − ˜ U $^{(0)}$, (5.28) ˜ U (0) = 1 2 ( π 2 1 + π 2 φ + φ ′ $^{2}$) − A$_{0}$ [ π ′ 1 + 1 2 q $^{2}$( a − 1) A$_{0}$ + q $^{2}$A$_{1}$ + qπ$_{φ}$ + qφ $^{′}$] − A$_{1}$ [ − qπ$_{φ}$ − 1 2 q $^{2}$( a + 1) A$_{1}$ − qφ $^{′}$] − Ω$_{1}$ θ + q $^{2}$( a − 1) θA ′ 1 − q $^{2}$θπ$_{1}$ − 1 2 q $^{2}$( a − 1) θ ′ 2 + 1 2 q $^{4}$θ2 − 1 2 q $^{2}$( a − 1) θ $^{2}$. (5.29) straint. Consequently, the model has a symmetry and this zero-mode is the generator of the infinitesimal gauge The contraction of the zero-mode ¯ ν (0) with the new sym- plectic potential above does not produce a new con- with the new symplectic potential Hence, the first-order Lagrangian (5.24) becomes indicating that it is necessary to obtain the rest of the cor- rection terms G ( n ) as functions of θ . This is accomplished just imposing that no more constraints are generated by the contraction of the zero-mode with the gradient of ex- tended symplectic potential. It allows us to determine the second order correction term G (2) as, The Lagrangian in (5.24) is not yet gauge invariant be- cause the zero-mode ¯ ν (0) still generates new constraints, given by with