14 action with respect to the Lagrange multiplier field λ ( x ) allows a solution for nonzero constant λ , i.e a finite Z mass. The equation which determines the mass is the one-loop tadpole condition that enforces 〈 Z † Z 〉 = 1, where M 2 ≡ γ $^{2}$λ/N . Introducing a momentum space cutoff Λ in the loop integral, we get a positive mass squared This result is a familiar example of dimensional transmutation, in which the physical scale is related to the cutoff scale by asymptotic freedom, as is the case in 4D QCD. A similar result is obtained in the string-theoretic cutoff scheme, where M 2 is given by (31), Thus the tadpole condition (32) determines the strength of the boundary perturbation describing the physical vacuum. The marginal boundary perturbation (1) is seen to be a generalization of the nonzero VEV acquired in the large N approximation by the Lagrange multiplier field λ ( x ). However, in the string theory case λ is not a constant, but is rapidly oscillating in one spatial direction at the cutoff scale. In the continuum limit, this rapidly oscillating field averages out to a net positive Z -particle mass shift. This is due to the fact that the Z -particle density also oscillates at the same rate, with the Z ’s living preferentially in the vicinity of the D-branes. From Eq. (30) we see that the effect of the boundary perturbation on the lowest lying Z band is an overall positive mass shift, just like the effect of the constant λ in the large N calculation. The elegance of the particular marginal perturbation (1) is that the mass shift appears through a shift of the Kac-Moody charge of the open string, thereby preserving world sheet conformal symmetry. To summarize the main result of this section, the effect of the marginal boundary perturbation in the string-theoretic formulation of the CP $^{N}$− 1 model is to induce a uniform shift in the Kac- Moody charges of the string states in the compactified R = R$_{0}$ theory, which is seen as a mass shift of the states in the R = ∞ theory. For the lowest lying band, the shift to a near-critical value of g eliminates the tachyonic instability. This shift represents the transition from the perturbative vacuum to the physical vacuum. The critical boundary perturbation g = 1 2 changes the boundary condition on the string from Neumann to Dirichlet, and corresponds to a periodic background of codimension-one D-branes [14, 15]. This picture of the physical vacuum nicely coincides with the codimension one, laminated topological charge structure observed in Monte Carlo simulations. 1 = − iγ 2 ∫ d $^{2}$p (2 π ) 2 1 p 2 + M 2 (32) M 2 = Λ $^{2}$e − 4 π/γ 2 (33)