Classical solutions of the TEK model and noncommutative instantons in two dimensions

Abstract

The twisted Eguchi-Kawai (TEK) model provides a non-perturbative de¯nition of noncommutative Yang-Mills theory: the continuum limit is approached at large-N by performing suitable double scaling limits, in which non-planar contributions are no longer suppressed. We consider here the two-dimensional case, trying to recover within this frame- work the exact results recently obtained by means of Morita equivalence. We present a rather explicit construction of classical gauge theories on noncommutative toroidal lattice for general topological charges. After discussing the limiting procedures to recover the theory on the noncommutative torus and on the noncommutative plane, we focus our at- tention on the classical solutions of the related TEK models. We solve the equations of motion and we ¯nd the con¯gurations having ¯nite action in the relevant double scaling limits. They can be explicitly described in terms of twist-eaters and they exactly cor- respond to the instanton solutions that are seen to dominate the partition function on the noncommutative torus. Fluxons on the noncommutative plane are recovered as well. We also discuss how the highly non-trivial structure of the exact partition function can emerge from a direct matrix model computation. The quantum consistency of the TEK formulation is eventually checked by computing Wilson loops in a particular limit