Doublet Groups, Extended Lie Algebras, and Well Defined Gauge Theories for the Two Form Field

TL;DR

This paper develops a gauge theory framework for two-form fields, extending Lie algebras, incorporating non-associative structures, and enabling minimal coupling with matter, thus broadening the understanding of Kalb-Ramond and related theories.

Contribution

It introduces a novel gauge symmetry for doublet form fields, extends Lie algebra structures, and formulates a gauge theory for the Kalb-Ramond field applicable to general groups.

Findings

Constructed a gauge connection including a two-form field.

Defined minimal coupling of the B-field with matter, revealing a conserved current.

Clarified the relation between topological Chern-Simons theories and B∧F theories.

Abstract

We propose a symmetry law for a doublet of different form fields, which resembles gauge transformations for matter fields. This may be done for general Lie groups, resulting in an extension of Lie algebras and group manifolds. It is also shown that non-associative algebras naturally appear in this formalism, which are briefly discussed. Afterwards, a general connection which includes a two-form field is settled-down, solving the problem of setting a gauge theory for the Kalb-Ramond field for generical groups. Topological Chern-Simons theories can also be defined in four dimensions, and this approach clarifies their relation to the so-called $B \wedge F$-theories. We also revise some standard aspects of Kalb-Ramond theories in view of these new perspectives. Since this gauge connection is built upon a pair of fields consisting of a one-form and a two-form, one may define Yang-Mills theories as usually and, remarkably, also {\it minimal coupling} with bosonic matter, where the Kalb-Ramond field appears naturally as mediator; so, a new associated conserved charge can be defined. For the Abelian case, we explicitly construct the minimal interaction between $B$-field and matter following a "gauge principle" and find a novel conserved tensor current. This is our most significative result from the physical viewpoint. This framework is also generalized in such a way that any $p$-rank tensor may be formulated as a gauge field.