13 Gauge invariance and dual equivalence of Abelian and non-Abelian actions via dual embedding formalism inal phase space variables and the WZ variable θ , into the first-order Lagrangian, right on the kinetic and sym- plectic potential sector, respectively. In this way, the first-order Lagrangian that governs the dynamics of the bosonized CSM, given in Eq. (5.2), is rewritten as This means that the model is not a gauge invariant the- ory. The gauge symmetry inside the model will be disclosed via a new gauge-invariant formalism that does not require more than one WZ field. The fundamental concept be- hind the symplectic gauge-invariant formalism dwells in the extension of the original phase space with the intro- duction of two arbitrary function Ψ( φ, π$_{φ}$, A$_{0}$, A$_{1}$, π$_{1}$, θ ) and G ( φ, π$_{φ}$, A$_{0}$, A$_{1}$, π$_{1}$, θ ), depending on both the orig- where The gauge-invariant formulation encompasses two steps: one is dedicated to the computation of Ψ while the other is addressed to the calculation of G . ˜ L (0) = π$_{φ}$ ˙ φ + π$_{1}$ ˙ A$_{1}$ + ˙ θ Ψ − ˜ U $^{(0)}$, (5.13) ˜ 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 $^{′}$] + G ( φ, π$_{φ}$, A$_{0}$, A$_{1}$, π$_{1}$, θ ) . (5.14) ˜ A (0) φ = π$_{φ}$, ˜ A (0) A$_{1}$ = π$_{1}$, ˜ A (0) A$_{0}$ = ˜ A (0) π$_{φ}$ = ˜ A (0) π$_{1}$ = 0 , (5.15) ˜ A (0) θ = Ψ . ˜ f (0) =     0 − 1 0 0 0 ∂ Ψ y ∂φ x 1 0 0 0 0 ∂ Ψ y ∂π x φ 0 0 0 0 0 ∂ Ψ y ∂A x 0 0 0 0 0 − 1 ∂ Ψ y ∂A x 1 0 0 0 1 0 ∂ Ψ y ∂π x 1 − ∂ Ψ x ∂φ y − ∂ Ψ x ∂π y φ − ∂ Ψ x ∂A y 0 − ∂ Ψ x ∂A y 1 − ∂ Ψ x ∂π y 1 f$_{θ}$$_{x}$$_{θ}$$_{y}$    $^{δ}$$^{(}$$^{x}$ − y ) , (5.16) The corresponding symplectic matrix ˜ f (0) reads The enlarged symplectic variables are now ˜ ξ (0) α = ( φ, π$_{φ}$, A$_{0}$, A$_{1}$, π$_{1}$, θ ) with the following one-form canoni- cal momenta From this relation a set of equations will be obtained and consequently, the arbitrary function Ψ can be deter- mined. We will now investigate the symmetry related to the following zero-mode, with bar representing a transpose matrix. To start, we multiply the zero-mode (5.19) by the sym- plectic matrix (5.16), as shown in equation (5.18). Due to this, some equations arise and after an integration Ψ is determined as To unveil the gauge symmetry hidden inside the model, we assume that this singular matrix has a zero-mode ( ν $^{(0)}$) that satisfies the following relation, with θ$_{x}$ ≡ θ ( x ), θ$_{y}$ ≡ θ ( y ), Ψ$_{x}$ ≡ Ψ( x ) and Ψ$_{y}$ ≡ Ψ( y ). Note that this matrix is singular since ∂ Ψ x ∂A y 0 = 0. Due to this, we conclude that Ψ ≡ Ψ( φ, π$_{φ}$, A$_{1}$, π$_{1}$, θ ). where f$_{θ}$$_{x}$$_{θ}$$_{y}$ = ∂ Ψ$_{y}$ ∂θ$_{x}$ − ∂ Ψ$_{x}$ ∂θ$_{y}$ , (5.17) ∫ ν (0) α ( x ) ˜ f (0) $_{αβ}$( x − y ) d y = 0 . (5.18) Ψ = π ′ + qφ ′ + qπ$_{φ}$ + q $^{2}$A$_{1}$ . (5.20)