Gauge Dependence in Chern–simons Theory

Abstract

We compute the contribution to the modulus of the one-loop effective action in pure non-Abelian Chern-Simons theory in an arbitrary covariant gauge. We find that the results are dependent on both the gauge parameter (α) and the metric required in the gauge fixing. A contribution arises that has not been previously encountered; it is of the form (α/ p 2 )ǫ µλν p λ . This is possible as in three dimensions α is dimensionful. A variant of proper time regularization is used to render these integrals well behaved (although no divergences occur when the regularization is turned off at the end of the calculation). Since the original Lagrangian is unaltered in this approach, no symmetries of the classical theory are explicitly broken and ǫ µλν is handled unambiguously since the system is three dimensional at all stages of the calculation. The results are *

Operator regularization and noncommutative Chern–Simons theory

“…From (15), (31) and (32), it is evident that the contribution to (30) that is bilinear in the background field A µ is (upon following the approach used above with the ζ-function)…”

“…The modulus and phase of the functional determinant associated with the effective action at one loop order are associated with the ζ-function [8,9] and η-function [1,[10][11][12] respectively. This approach has been used in conventional Chern Simons theory [11][12][13][14][15].…”