16 FIG. 3: One loop diagrams contributing to the large N effective gauge potential in CP N$_{−}$ $^{1}$. 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). 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 with χ$_{t}$ = 3 M $^{2}$/πN . This corresponds to a one-loop contribution to the effective gauge potential, 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 L eff = − N 48 π $_{2}$F 2 µν (37) Π$_{µν}$ = 1 4 π $^{2}$χ$_{t}$ ( q $^{2}$g$_{µν}$ − q$_{µ}$q$_{ν}$ ) (36) V. THE EFFECTIVE GAUGE ACTION: TACHYONIC CRYSTALS AND QUARK CONFINEMENT + 