content_image/337203.jpg content_image/337204.jpg content_image/337205.jpg content_image/337206.jpg content_image/337207.jpg
will use this connection in Section V to obtain the effective space-time gauge action and compare it with the large N result. 

From the expression (7) for the two-point function we determine the path- and field-dependent action for Z -particle propagation in the CP $^{N}$− 1 model to be (up to an overall constant which determines the scale of the proper time parameter τ ) 

<img src='content_image/337203.jpg'>

The string theoretic cutoff procedure described above consists of identifying the path action (8) as the l → 0 limit of the worldsheet action for an open bosonic string coordinate X ( σ, τ ), 

<img src='content_image/337204.jpg'>

The bulk term is the usual CFT for a bosonic string coordinate, defined inside a rectangular area R of width l and length T in the complex z ≡ τ + iσ plane. 

<img src='content_image/337205.jpg'>

The worldsheet action also includes two boundary terms, corresponding to the spacetime fields λ ( X ) and A$_{µ}$ ( X ) interacting with one or both ends of the string, The most physically sensible formulation for our purposes is to put the Z -particle charge that interacts with the background fields at one end of the string ( σ = 0) and impose Neumann boundary conditions at the other end ( σ = l ). (Physically, the charged particle still follows a definite path x$_{µ}$ ( τ ) = X$_{µ}$ (0 , τ ) but now has a string attached to it.) Thus we take the boundary term to be along the real axis, 

<img src='content_image/337206.jpg'>

(9)-(11) defines the worldsheet action for the coordinates of an open bosonic string in background tachyon and U(1) gauge fields. 

The string theoretic cutoff for the CP $^{N}$− 1 model may also be understood by writing the one-loop vacuum amplitude for an open string in a spectral representation which sums over intermediate open string states. This spectral analysis also exposes the essential role of the open string tachyon, and its connection to the instability of the λ = 0 vacuum seen in the large N field theory. Let us begin by considering the one-loop vacuum string amplitude G ( x, x ) without background fields (i.e. λ = A$_{µ}$ = 0). This amplitude has the form 

<img src='content_image/337207.jpg'>

