Smooth Gauge Strings and D>=2 Lattice Yang-Mills Theories

TL;DR

This paper derives a gauge string representation for D>=3 lattice Yang-Mills theories in the strong coupling phase, establishing a duality that predicts confinement and connects to the AdS/CFT correspondence.

Contribution

It generalizes the Gross-Taylor string reformulation to higher dimensions and develops an algorithm to determine weights for YM-flux worldsheets in D>=3 lattice theories.

Findings

Establishes a YM/String duality in D>=2.

Predicts confinement in large N strong coupling regime.

Connects lattice YM theories to continuous YM models via smooth gauge strings.

Abstract

Employing the nonabelian duality transformation, I derive the Gauge String form of certain D>=3 lattice Yang-Mills (YM_{D}) theories in the strong coupling (SC) phase. With the judicious choice of the actions, in D>=3 our construction generalizes the Gross-Taylor stringy reformulation of the continuous YM_{2} on a 2d manifold. Using the Eguchi-Kawai model as an example, we develope the algorithm to determine the weights w[\tilde{M}] for connected YM-flux worldsheets \tilde{M} immersed into the 2d skeleton of a D>=3 base-lattice. Owing to the invariance of w[\tilde{M}] under a continuous group of area-preserving worldsheet homeomorphism, the set of the weights {w[\tilde{M}]} can be used to define the theory of the smooth YM-fluxes which unambiguously refers to a particular continuous YM_{D} system. I argue that the latter YM_{D} models (with a finite ultraviolet cut-off) for sufficiently large bare coupling constant(s) are reproduced, to all orders in 1/N, by the smooth Gauge String thus associated. The asserted YM_{D}/String duality allows to make a concrete prediction for the 'bare' string tension \sigma_{0} which implies that (in the large N SC regime) the continuous YM_{D} systems exhibit confinement for $D\geq{2}$. The resulting pattern is qualitatively consistent (in the extreme D=4 SC limit) with the Witten's proposal motivated by the AdS/CFT correspondence.