On the quantum equivalence of commutative and noncommutative Chern-Simons theories at higher orders

TL;DR

This paper demonstrates the quantum equivalence of commutative and noncommutative Chern-Simons theories at two-loop order by showing cancellations in quantum corrections to specific Wilson line correlators, supporting the conjecture of their perturbative equivalence.

Contribution

It provides the first explicit two-loop calculation of correlation functions in noncommutative Chern-Simons theory, confirming the quantum equivalence with the commutative case at this order.

Findings

Two-loop quantum corrections cancel out, supporting equivalence.

Cancellations occur in gauge-invariant correlators involving Wilson lines.

Speculation on all-order results for these correlators.

Abstract

We continue our investigation of the quantum equivalence between commutative and noncommutative Chern-Simons theories by computing the complete set of two-loop quantum corrections to the correlation function of a pure open Wilson line and an open Wilson line with a field strength insertion, on the noncommutative side in a covariant gauge. The conjectured perturbative equivalence between the free commutative theory and the apparently interacting noncommutative one requires that the sum of these corrections vanish, and herein we exhibit the remarkable cancellations that enforce this. From this computation we speculate on the form of a possible all-order result for this simplest nonvanishing correlator of gauge invariant observables.