content_image/334511.jpg content_image/334509.jpg content_image/334508.jpg
<img src='content_image/334511.jpg'>

be reinterpreted as infrared divergences in the dual channel. In the next section, we will investigate the effective action of the gauge field A$_{µ}$ coupled to the open string as in Eq. (11). 

## V. THE EFFECTIVE GAUGE ACTION: TACHYONIC CRYSTALS AND QUARK CONFINEMENT 

A central feature of the large N solution in CP $^{N}$− 1 is the appearance of a dynamically generated linear confining Coulomb potential between charged Z -particles. This arises from the fact that, although the bare Lagrangian does not include an F 2 term, such a term appears in the effective gauge potential at the one-loop level. This is seen by computing the one-loop photon self energy given by the diagrams in Fig. 3 (with Z -propagators of mass M ). In the limit q 2 → 0, this gives a self-energy tensor of the form 

<img src='content_image/334509.jpg'>

with χ$_{t}$ = 3 M $^{2}$/πN . This corresponds to a one-loop contribution to the effective gauge potential, 

<img src='content_image/334508.jpg'>

This promotes the auxiliary gauge field A$_{µ}$ to a full dynamical Maxwell field, and in particular, leads to a linear confining potential between charged Z ’s. 

In this Section we discuss the effective gauge potential in the string theoretic cutoff theory. Thus we consider an open string coupled to a background abelian gauge field by the Wilson line interaction, Eq. (11). This problem has been studied in the context of boundary conformal field theory in Ref. [19]. To present a self-contained discussion, we briefly summarize the results needed to obtain the effective gauge potential for the CP $^{N}$− 1 model. The analysis of Ref. [19] relies on an 

