Open Wilson lines and group theory of noncommutative Yang-Mills theory in two dimensions

Abstract

The correlation functions of open Wilson line operators in two-dimensional Yang-Mills theory on the noncommutative torus are computed exactly. The correlators are expressed in two equivalent forms. An instanton expansion involves only topological numbers of Heisenberg modules and enables extraction of the weak-coupling limit of the gauge theory. A dual algebraic expansion involves only group theoretic quantities, winding numbers and translational zero modes, and enables analysis of the strong-coupling limit of the gauge theory and the high-momentum behaviour of open Wilson lines. The dual expressions can be interpreted physically as exact sums over contributions from virtual electric dipole quanta.

Instantons, fluxons, and open gauge string theory

“…The simplest way to understand this stringiness is through the strong coupling expansion of the noncommutative gauge theory, which may be expressed as a sum over non-local electric dipole configurations [13]. In this sense the instanton expansion is dual to an open string representation of the gauge theory, in which the endpoints of strings are coupled to a background magnetic field B = (A θ/2π) −1 and the strings become polarized as dipoles.…”

“…Note thatg 2 c → ∞ in the commutative limit θ → 0. This would then suggest that the physics beyond the transition point is accurately described by the strong coupling expansion which can be expressed in terms of contributions from the production of virtual electric dipoles in the vacuum [13]. In the commutative case this would be exactly the regime in which a string representation of the gauge theory is available.…”

“…However, this string representation is based on the strong-coupling expansion of the commutative gauge theory. As we mentioned earlier, while such an expansion is available for the gauge theory on the noncommutative torus [13], it is rather complicated and does not straightforwardly admit nice algebraic or geometric characterizations. We must therefore try to extract the string representation of two-dimensional noncommutative Yang-Mills theory directly from its weak-coupling expansion.…”

“…These aspects become particularly interesting in two dimensions, because ordinary Yang-Mills theory on a Riemann surface has a very precise interpretation as a string theory [3]- [5], while the exact solution of gauge theory on the noncommutative torus T 2 θ has been recently presented through an expansion in noncommutative instantons in [12,13]. An explicit realization of two-dimensional noncommutative gauge theory as a matrix model has also been constructed in [14,15] by exploiting the general relation [16] between its lattice regularization and the twisted Eguchi-Kawai (TEK) model [17]- [19].…”

Localization for Yang-Mills Theory on the Fuzzy Sphere

“…Our model fills this gap, providing a gauge theory on the fuzzy sphere whose exact solution is on a unified footing with that of gauge theory on the noncommutative torus, in the same way that all two-dimensional gauge theories admit universal solutions. This is even apparent from the strong coupling expansions of the two noncommutative gauge theories [33,34], which exhibit the same degrees of complexity.…”