Classical Solutions of the TEK Model and Noncommutative Instantons in Two Dimensions

TL;DR

This paper explores classical solutions in the two-dimensional TEK model, revealing instanton configurations and fluxons that match known noncommutative gauge theory results, and discusses their implications for nonperturbative formulations.

Contribution

It provides explicit classical solutions of the TEK model in two dimensions, connecting them to noncommutative instantons and fluxons, and analyzes the partition function structure.

Findings

Classical solutions correspond to noncommutative instantons and fluxons.

Explicit construction of gauge theories on noncommutative tori with various topological charges.

The partition function's structure can be derived from matrix model computations.

Abstract

The twisted Eguchi-Kawai (TEK) model provides a non-perturbative definition 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 framework 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 attention on the classical solutions of the related TEK models. We solve the equations of motion and we find the configurations having finite action in the relevant double scaling limits. They can be explicitly described in terms of twist-eaters and they exactly correspond 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.