A$_{µ}$ = Z † ( ∂$_{µ}$Z ) − ( ∂$_{µ}$Z ) † Z (3) 6 is D$_{µ}$ = ∂$_{µ}$ − iA$_{µ}$ . Note that there is no ( F$_{µν}$ ) 2 term for the gauge field, so A$_{µ}$ is a dummy field whose equation of motion sets it equal to the U (1) current, If the A$_{µ}$ field is integrated out, the theory can be written as a nonlinear sigma model of self- interacting Z ’s. Because of the absence of a kinetic term for the gauge field, the model is classically conformally invariant. It acquires a mass scale via dimensional transmutation, as in 4-dimensional QCD. An ( F$_{µν}$ ) 2 term for the gauge field is generated by loop effects, as can be seen explicitly in the large N solution [16, 17]. This produces a linear Coulomb potential which confines U (1) charge. The generation of a kinetic term in the effective gauge action is a direct consequence of the dynamical generation of a mass for the Z -particles. In large N, the transition from the perturbative to the physical vacuum is constrained by a one-loop consistency condition which determines the “constituent” mass of the Z particles. In our string-cutoff theory, we will show that this transition is described in terms of the marginal boundary perturbation (1) in the BCFT for a massless scalar field on the string world sheet. To clarify the connection between the CP $^{N}$− 1 model and boundary CFT, it is useful to introduce a proper time formalism. First let us impose the constraint Z † Z = 1 via an auxilary Lagrange multiplier field, Thus the model reduces to charged scalar particles interacting with a real scalar field λ ( x ) and a vector field A$_{µ}$ ( x ). We will focus on the two-point function for Z particles, By introducing a proper time variable, we can replace the path integral over the Z fields by an integral over spacetime particle paths x $^{µ}$( τ ), µ = 1 , 2. (We work in two-dimensional Euclidean space.) The exact propagator for a charged particle with fixed λ and A$_{µ}$ fields is then given in a proper time representation, where the Green’s function for the proper time Schrodinger equation is given by summing over particle paths with boundary conditions x $^{µ}$(0) = x$_{1}$, x $^{µ}$( T ) = x$_{2}$ . The interaction with the gauge field is given by the Wilson line along each path, ˆ G ( x$_{1}$, x$_{2}$, T ) = ∫ D x exp i [ ∫ T 0 dτ ( ( dx µ dτ ) 2 − λ ( x ) − ∫ dτ dx µ dτ A$_{µ}$ ( x ) ] (7) G ( x$_{1}$, x$_{2}$ ) = ∫ ∞ 0 ˆ G ( x$_{1}$, x$_{2}$, T ) dT (6) 〈 Z$_{i}$ ( x$_{1}$ ) $_{Z}$† $_{j}$( x$_{2}$ ) 〉 = 1 N δ$_{ij}$G ( x$_{1}$, x$_{2}$ ) (5) L → L − λ ( x )( Z † Z − 1) (4)