content_image/382427.jpg content_image/382428.jpg content_image/382429.jpg content_image/382430.jpg content_image/382439.jpg content_image/382440.jpg content_image/382441.jpg
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- 

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 

where 

<img src='content_image/382427.jpg'>

<img src='content_image/382428.jpg'>

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 . 

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 

The corresponding symplectic matrix ˜ f (0) reads 

<img src='content_image/382429.jpg'>

<img src='content_image/382430.jpg'>

where 

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 ). 

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, 

<img src='content_image/382439.jpg'>

<img src='content_image/382440.jpg'>

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 

<img src='content_image/382441.jpg'>

