content_image/332686.jpg content_image/332687.jpg
can be obtained from the α ′ → 0 limit of a bosonic open string theory with one end interacting with a background gauge field A$_{µ}$ ( X ) and tachyon field λ ( X ). In this formulation of the model, the string length √$_{α}$$_{′}$ serves as a short distance cutoff and plays a role analogous to the lattice spacing in the lattice formulation of the cutoff theory. With this string theoretic cutoff, we can relate the dynamics of the CP $^{N}$− 1 model to that of a boundary conformal field theory (BCFT), where the tachyon background λ ( X ) admits an exactly marginal, oscillatory boundary perturbation of the form [11, 13] 

<img src='content_image/332686.jpg'>

Here X (0) is the endpoint coordinate of an open bosonic string, given by the boundary value on the real axis of holomorphic and antiholomorphic conformal fields in the upper half plane. The phase of the parameter g fixes the overall position of the oscillating field along the X axis. From here on we will take g to be real and positive. Sen has shown that this boundary perturbation (1) represents the Euclidean world sheet view of the homogeneous decay of an unstable D-brane [12]. The theory without a boundary term ( g = 0) corresponds to Neumann boundary conditions on the X coordinate, i.e. a homogeneous D-brane spanning that coordinate. For 0 < g < 1 $_{2}$, the perturbation (1) is marginal and belongs to an SU(2) current algebra. As we discuss in the following Section, this boundary deformation preserves world sheet conformal symmetry, and the open string spectrum is obtained in terms of U (1) Kac-Moody modules of a “phase shifted” Sugawara Hamiltonian. At a critical value g = 1 2 the b.c.’s on X become pure Dirichlet conditions. In space-time, this represents the background state reached after the brane decay process is complete, consisting of an alternating-sign codimension-one array of D-branes arranged along the X direction in a one- dimensional tachyonic crystal [14, 15]. 

## II. STRING THEORY AS AN ULTRAVIOLET COMPLETION OF THE CP N$_{−}$ 1 MODEL 

The CP $^{N}$− 1 model is a nonlinear sigma model of N self-interacting complex scalar fields Z$_{i}$ ( x ) , i = 1 ,... ,N with the constraint Z † Z = 1. The most convenient formulation of the model for our purposes is obtained by introducing an auxiliary abelian gauge field A$_{µ}$ ( x ). Then the action consists solely of a gauge covariant kinetic term, 

<img src='content_image/332687.jpg'>

Here we will denote the CP $^{N}$− 1 coupling constant by γ (rather than the more conventional g ) to avoid confusion with the coefficient of the boundary perturbation. In (2) the covariant derivative 

