18 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism Integrating this relation in θ $^{f}$( w ), the quadratic correc- tion term is obtained as Consequently, the first-order Lagrangian is rewritten as where the symplectic potential is It completes the first step of the symplectic gauge- invariant formulation. To unveil the hidden symmetry inside the model, the zero-mode ¯ ν (0) does not generate new constraints, conse- quently, we have the following relation, where In this way, two correction terms as functions of θ$_{a}$ ( G $^{(3)}$( x ) and G $^{(4)}$( x )) remain to be computed. Let us compute the first one. It can be done from the following relation, From this relation we can compute the whole set of cor- rection terms as functions of θ $^{a}$. The linear correction term in θ is computed as being, After an integration we have that and we can write that, Now, we will compute the quadratic term, namely, Finally, the last correction term is, Therefore, the gauge invariant first-order Lagrangian is where the gauge invariant Hamiltonian, identified as be- ing the symplectic potential, is given by ˜ L (0) = π i a ˙ A a i − ( ∂$_{i}$π i $_{a}$) ˙ θ a − ˜ V $^{(0)}$, (6.27) G $^{(1)}$( x ) = − gC b $_{fa}$∂ x i ( A a $_{0}$( x ) π i $_{b}$( x )) θ $^{f}$( x ) − m $^{2}$( ∂ x i A i $_{f}$) θ $^{f}$( x ) − 1 2 ∫ y ∂ y i ( F a $_{kj}$( x ) ∂F kj a ( x ) ∂A f i ( y ) ) θ $^{f}$( y ) . (6.21) G $^{(3)}$( x ) = − 1 2 ∫ θ $^{g}$( z ) ∫ z ∂ z n ∫ θ$_{f}$ ( x ) ∫ w ∂ w k ∫ y ∂ y l ∂ A kl fb ∂A n $_{g}$( z ) θ $^{b}$( y ) . (6.25) ∫ x { ∂ w i ∂V ( x ) (0) ∂A f i ( w ) + G $^{(1)}$( x ) ∂θ $^{f}$( w ) } = 0 . (6.20) ∫ z { ∂ z n [ 1 2 ∫ θ $^{f}$( x ) ∫ w ∂ w k ∫ y ∂ y l ∂ A kl fb ∂A g $_{n}$( z ) θ $^{b}$( y ) ] + G $^{(3)}$( x ) ∂θ$_{g}$ ( z ) } = 0 ∫ x ¯ ν (0) α ( w ) ∂ ˜ V ( x ) ∂ξ a $_{α}$( w ) = 0 . (6.19) A il fb = ∂F a $_{kj}$( x ) ∂A f i ( w ) ∂F kj a ( x ) ∂A b $_{l}$( y ) + F a $_{kj}$( x ) ∂ $^{2}$F kj a ( x ) ∂A f i ( w ) ∂A b $_{l}$( y ) . (6.24) G $^{(2)}$( x ) = 1 2 m $^{2}$( ∂ i $_{x}$θ$^{f}$( x )) 2 + 1 2 ∫ θ $^{f}$( x ) ∫ w ∂ w i ∫ y [ ( ∂ y l A il $_{fb}$) θ $^{b}$( y ) ] , (6.23) ˜ V (0) = 1 2 ( π i $_{a}$) 2 − A a $_{0}$Ω$_{a}$ + 1 2 m $^{2}$A a i A i a + 1 2 m $^{2}$A a $_{0}$A 0 a + 1 4 F a $_{kj}$F a kj + G. (6.18) ˜ L (0) = π i a ˙ A a i − ( ∂$_{i}$π i $_{a}$) ˙ θ a − ˜ V $^{(0)}$, (6.17) Ψ$_{a}$ = − ∂$_{i}$π i $_{a}$( x ) . (6.16) ∫ x { ∂ w i ∂ G $^{(1)}$( x ) ∂A f i ( w ) + G $^{(2)}$( x ) ∂θ $^{f}$( w ) } = 0 . (6.22) G $^{(4)}$( x ) = 1 2 ∫ θ $^{h}$( v ) ∫ v ∂ v i ∫ θ$_{g}$ ( z ) ∫ z ∂ z n ∫ θ$_{f}$ ( x ) ∫ w ∂ w k ∫ y ∂ y l ∂ $^{2}$A kl fb ∂A i $_{h}$( v ) ∂A n $_{g}$( z ) θ $^{b}$( y ) . (6.26)