Wilson loops and area-preserving diffeomorphisms in twisted noncommutative gauge theory

Abstract

We use twist deformation techniques to analyse the behaviour under area-preserving diffeomorphisms of quantum averages of Wilson loops in Yang-Mills theory on the noncommutative plane. We find that while the classical gauge theory is manifestly twist covariant, the holonomy operators break the quantum implementation of the twisted symmetry in the usual formal definition of the twisted quantum field theory. These results are deduced by analysing general criteria which guarantee twist invariance of noncommutative quantum field theories. From this a number of general results are also obtained, such as the twisted symplectic invariance of noncommutative scalar quantum field theories with polynomial interactions and the existence of a large class of holonomy operators with both twisted gauge covariance and twisted symplectic invariance.

Area-preserving diffeomorphisms in gauge theory on a non-commutative plane: a lattice study

“…Perturbation theory can reveal this effect, depending on the order considered. Since this is still restrictive, to a finite order the symmetry may be protected in part [11], with a residual subgroup showing up in other approaches as well [13].…”

“…Recently, ref. [13] reconsidered this issue by applying twist deformation techniques, which also confirm that the APD symmetry may break at the quantum level. A new study of the large θ expansion [14] reports the breakdown of the area law at O(1/θ 2 ), in agreement with simulations results at finite θ [15].…”

Wilson loops in 2D noncommutative euclidean gauge theory: 2. 1/θ expansion

“…As well as in the θ = 0 case, investigation of non-perturbative effects in a low-dimensional model is expected to prepare us for the analysis of a more complicated four-dimensional quantum dynamics. An incomplete list of papers, devoted to this direction of research, is presented in references [3]- [23].…”

Symmetries and Renormalization of Noncommutative Field Theory

“…One generically has ∆ θ (X ) = ∆ 0 (X ), but one can construct a "twisted" tensor calculus such that star-products of tensor fields transform as tensors. However, only unimodular U (∞) twisted transformations, with ∂ i X i = 0, preserve action functionals of noncommutative field theories [33].…”

“…One resolution, proposed in [36] (see also [33]), is to use a proper covariant noncommutative differential calculus to perform the Noether analysis relating fields and conserved charges. However, the physical origins of these symmetries are unclear, and in particular string theory is unable to account for twisted diffeomorphisms [37].…”