Twistor Wilson loops in large-$N$ Yang-Mills theory

TL;DR

This work analyzes twistor Wilson loops in large-$N$ Yang-Mills theory within a noncommutative spacetime framework. By constructing twistor Wilson loops in both operator and coordinate representations and leveraging the Groenewold-Moyal product, the authors show that the planar vacuum expectation value of these loops equals unity, $ig\langle W_\lambda(C_{u\bar{u}}) \big\rangle_{pl} = 1$, to all orders in perturbation theory and at leading $N$. The proof rests on Euclidean invariance, a special symmetry in the noncommutative setup, and an appropriate regulator to handle singularities, mirroring mechanisms from SUSY Wilson loops. This result supports the idea that a topological field theory sector underlies large-$N$ YM at leading order and sets the stage for extending to nonperturbative information on glueballs via a Chern-Simons–type construction on noncommutative spacetime.

Abstract

It has been known for many years that, in Yang-Mills theories with $\mathcal{N}=4,2,2^*$ supersymmetry, certain nontrivial supersymmetric Wilson loops exist with v.e.v. either trivial or computable by localization that arises from a cohomological field theory, which also computes the nonperturbative prepotential in $\mathcal{N}=2,2^*$ theories. Moreover, some years ago it has been argued that, in analogy with the supersymmetric case, certain nontrivial twistor Wilson loops with trivial v.e.v. to the leading large-$N$ order exist in pure SU($N$) Yang-Mills theory and are computed, to the leading large-$N$ order, by a topological field/string theory that, to the next-to-leading $\frac{1}{N}$ order, conjecturally captures nonperturbative information on the glueball spectrum and glueball one-loop effective action as well. In fact, independently of the above, it has also been claimed that "every gauge theory with a mass gap should contain a possibly trivial topological field theory in the infrared", so that the aforementioned twistor Wilson loops realize a stronger version of this idea, as they have trivial v.e.v. at all energy scales and not only in the infrared. In the present paper, we provide a detailed proof of the triviality of the v.e.v. of twistor Wilson loops at the leading large-$N$ order in Yang-Mills theory that has previously been only sketched, opening the way to further developments.