6 Discussion

Recent results about $UV/IR$ interplay, in the framework of noncommutative Yang-Mills theories [15], suggest to investigate the behavior of the action (36) by changing the size of the quantization volume $V$. In the small volume limit, i.e. $V << 1/ 4g^2\, M^4$, the main contribution to $S^{(\mathrm{q})}_{\mathrm{BI\, }}$ comes from the $\mathrm{det}[\,\gamma_{mn}\,]$ and the Nambu-Goto action is recovered again. The quantization volume drops out and the string tension, i.e. $\mu_0=M^2/4\pi g$, is determined by the only relevant mass scale $M$.
In the opposite, large volume, limit, i.e. $V >> 1/ 4g^2\, M^4$, the second term in the square root is small with respect to $1$ and the first non-vanishing contribution of the Taylor expansion is the Schild action. In this regime the relevant energy scale is set $V^{-1/4}$ and the corresponding string tension is $\mu_0=1/32\pi g^2\sqrt V$.
From a different point of view, Fairlie has recently pointed out some intriguing analogy between the Born-Infeld and Nambu-Goto actions [14]. Our results supports this connection. We believe we bridged the gap between four dimensional gauge theories and fully reparametrization invariant string models. We found new, non-trivial, relationship between a class of generalized $SU(\, N\,)$ models and Born-Infeld/Nambu-Goto strings, provided a new matrix degree of freedom, $\mathbf{\Phi}$, is introduced. In the original $SU(\, N\,)$ model $\mathbf{\Phi}$ connects the Yang-Mills phase, at $\mathbf{\Phi}= \mathbf{I}$, with the, non-linear, Born-Infeld phase, where the eigenvalues $\phi_{(a)}$ are given by equation (18). In the large-$N$ limit the order parameter becomes a world-sheet auxiliary field linearizing a square root type Born-Infeld string model. The small, respectively, large volume limits of this model correspond to Nambu-Goto and Schild actions.