Open Wilson lines and generalized star product in noncommutative scalar field theories

Abstract

Open Wilson line operators and generalized star product have been studied extensively in noncommutative gauge theories. We show that they also show up in noncommutative scalar field theories as universal structures. We first point out that dipole picture of noncommutative geometry provides an intuitive argument for robustness of the open Wilson lines and generalized star products therein. We calculate one-loop effective action of noncommutative scalar field theory with cubic self-interaction and show explicitly that the generalized star products arise in the nonplanar part. It is shown that, at low-energy, large noncommutativity limit, the nonplanar part is expressible solely in terms of the scalar open Wilson line operator and descendants.

Anatomy of one-loop effective action in non-commutative scalar field theories

“…In particular, we were able to show that, at large noncommutativity limit, the nonplanar contribution is expressible in terms of open Wilson line operators, thus completing the proof of our earlier result in [1]. The planar contribution, on the other hand, gives rise to a noncommutative version of the Coleman-Weinberg type potential, in contrast to the anticipation that the planar part ought to be expressible in terms of Wilson loop opeators -the putative master field in planar limit of matrix-valued field theories.…”

“…As such, the dipoles are ubiquitous to any noncommutative field theory, an aspect which would explain why the open Wilson line operators play prominent roles, not only in gauge theories but also in scalar field theories, in which neither gauge invariance nor gauge field is present. In [1], the effective action was calculated via the standard Feynman diagrammatics. To supplement the method, in this paper, we will be computing the effective action in the worldline method extended to noncommutative field theories, and make detailed comparison, wherever possible, with methods and results in the standard Feynman diagrammatics as well as in the point-particle limit of the open string worldsheet computation.…”

Strings, dipoles and fuzzy spheres

“…Thus I have shown that the vertex interaction of dipoles is naturally described using traces and matrix multiplication, the latter being equivalent to the * -multiplication of the fuzzy sphere algebra (14). This is the main result of the paper.…”

“…It was emphasized in [12] that at low energy strings behave as dipoles with the charges in the LLL and the joining and separation interaction of these dipoles is naturally described by gauge theory on the noncommutative plane [13,12]. For further detailed studies see also [14,15,16,17].…”

Multipliers in weighted Sobolev spaces

“…Refs [39]. pointed out that the appearance of open Wilson lines in the effective action is a generic feature of field theories on NC geometry.…”