20 The issue of universality (i.e. whether different cutoffs lead to the same low energy physics) may be further illuminated in the CP $^{N}$− 1 model by detailed comparison between the string theory cutoff, the lattice cutoff, and the large N approximation with momentum space cutoff. In fact an apparent discrepency has been pointed out between the lattice formulation and the continuum large N approximation associated with the phenomenon of “superconfinement” [22, 23]. This is the observation that, to all orders in the lattice strong coupling expansion, the net U (1) current (3) flowing along any link of the lattice must be zero, since the integral over the gauge link vanishes unless there are an equal number of U ’s and U † ’s. This would seem to rule out bulk electric field excitations. What appears in the large N approximation as a constant electric field, e.g. between spatially separated test charges, when looked at microscopically on the lattice is a chain of polarized Z $^{+}$Z − pairs. The large N approximation thus gives a somewhat smeared out view of the short- distance lattice dynamics. On the other hand, lattice Monte Carlo calculations of χ$_{t}$ converge nicely with increasing N to be in accurate agreement with the large N result at about N ≈ 9 [4]. So the absence of superconfinement in the large N approximation does not seem to invalidate its low energy predictions. As discussed in [23] the smearing of the superconfinement constraint and the appearance of bulk electric fields in the large N analysis can be traced to the introduction of the Lagrange multiplier field λ , which temporarily relaxes the constraint Z † Z = 1. This constraint is reimposed upon integrating over the λ field, but for fixed λ , superconfinement does not hold. (Note that typically the Lagrange multiplier field is not introduced in lattice calculations and the constraint Z † Z = 1 is maintained site by site in the update procedure.) The large N approximation, in which λ is assumed to be a constant, is not sufficient to restore the superconfinement property. It is very interesting that the string theoretic cutoff introduced in this paper leads to a λ ( x ) field which is rapidly oscillating at the cutoff scale but has the same low energy effect as a constant λ , i.e. a constant mass shift. The tachyonic crystal structure for the gauge field which emerges from the string theory seems to be to some extent replicating the short-distance structure seen on the lattice. This work was supported in part by the Department of Energy under grant DE-FG02- 97ER41027. [1] I. Horvath et al, Phys. Rev. D68: 114505 (2003);. [2] I. Horvath et al, Phys. Lett. B612: 21 (2005);. [3] S. Ahmad, J. T. Lenaghan and H. B. Thacker, Phys. Rev. D72: 114511 (2005).