2-Form Gauge Field Theories and "No Go" for Yang-Mills Relativistic Actions

TL;DR

This paper investigates the gauge group structures of 2-form fields, revealing that standard Yang-Mills actions are not relativistically invariant for these fields unless they are Abelian, highlighting fundamental limitations in their physical formulation.

Contribution

It clarifies the group structures underlying 2-form gauge potentials and demonstrates the non-invariance of standard Yang-Mills actions for non-Abelian cases.

Findings

2-form gauge fields correspond to extensions of Lie groups

Standard Yang-Mills actions are not relativistically invariant for non-Abelian 2-form fields

A 2-form field can be viewed as a connection on a flat Euclidean manifold

Abstract

The transformation properties of a Kalb-Ramond field are those of a gauge potential. However, it is not clear what is the group structure to which these transformations are associated. In this paper, we complete a program started in previous articles in order to clarify this question. Using the spectral theorem, we improve and generalize previous approaches and find the possible group structures underneath the 2-form gauge potential as extensions of Lie groups, when its representations are assumed to act into any tensor (or spinor) space with inner product. We also obtain a fundamental representation where a two-form field turns out to be a connection on a flat Euclidean basis manifold, with a corresponding canonical curvature. However, we show that these objects are not associated to space-time tensors and, in particular, that a standard Yang-Mills action is not relativistically invariant, except (as expected) in the Abelian case. This is our main result, from the physical point of view.