[Paper Review] The Kalb-Ramond field as a connection on a flat space time

M. Botta Cantcheff|ArXiv.org|Dec 16, 2002

Quantum chaos and dynamical systems1 references3 citations

TL;DR

This paper proposes a geometric realization of the Abelian Kalb-Ramond field as a connection on flat Minkowski spacetime by constructing a Lie group via tensor products of gauge groups, where the connection naturally yields a rank-2 tensor field. The key result is that the field strength derived from this connection reproduces the standard Kalb-Ramond field strength $ H_{\mu\nu\rho} = 2\partial_{[\mu}B_{\nu]\rho} $, confirming its gauge invariance and consistency with string theory and topological field theories.

ABSTRACT

We obtain a (Abelian) two form field as a connection on a flat space-time and its corresponding field strength is canonically constructed.

Motivation & Objective

• To systematically derive the Kalb-Ramond field as a connection on flat spacetime from a group-theoretic construction.
• To resolve the issue of separability of the gauge parameter $ \beta^{a}_{\mu} = \beta^{a}v_{\mu} $ by using tensor products of Lie groups.
• To define a covariant derivative and field strength for the two-form connection that reproduce the standard Kalb-Ramond dynamics.
• To show that the antisymmetric part of the connection corresponds to the physical Kalb-Ramond field $ b_{\mu\nu} $, while the symmetric part is a redundant degree of freedom.
• To lay the foundation for non-Abelian extensions and coupling to fermionic matter in a geometrically consistent framework.

Proposed method

• Construct a Lie group $ \bar{G} $ as a tensor product of four copies of a gauge group $ G $, each associated with a spacetime basis vector $ dx^{\underline{\mu}} $, using Dirac matrices $ \gamma^{\mu} $ to encode the spacetime structure.
• Define the gauge parameter as $ \rho = \alpha^{a}I + \beta^{a}_{\mu}v_{\mu}\gamma^{\mu} $, where $ \beta^{a}_{\mu} $ is a one-form, and exponentiate to obtain group elements $ g = \exp[i(\alpha^{a} + \beta^{a}_{\mu}v_{\mu}\gamma^{\mu})\tau^{a}] $.
• Introduce a connection $ B_{\mu} = B_{\mu\nu}\Gamma^{\nu} $, where $ \Gamma^{\nu} $ are matrices formed from tensor products of Dirac matrices, ensuring the connection transforms as a two-form under gauge transformations.
• Define the covariant derivative $ \nabla_{\mu} = \partial_{\mu} - iB_{\mu} $, and derive the field strength via the commutator $ [\nabla_{\mu}, \nabla_{\nu}] = -i{\cal F}_{\mu\nu} $, yielding $ {\cal F}_{\mu\nu\rho} = 2\partial_{[\mu}B_{\nu]\rho} $.
• Identify the antisymmetric part $ b_{\mu\nu} = B_{[\mu\nu]} $ as the physical Kalb-Ramond field, and the totally antisymmetric part $ H_{\mu\nu\rho} = {\cal F}_{[\mu\nu\rho]} $ as the standard field strength.
• Use a Lorentz-covariant formulation with matrices $ \Gamma^{\mu}_{(i)} $ to ensure invariance under spacetime transformations, and show that the group parameter transforms as $ {\cal B} \to \sigma{\cal B}\sigma^{-1} $.

Experimental results

Research questions

• RQ1 Can the Kalb-Ramond field be systematically derived as a connection on flat spacetime via a group-theoretic construction?
• RQ2 How can the gauge parameter $ \beta_{\mu} $, which transforms as a one-form, be consistently embedded into a Lie group structure without requiring separability $ \beta^{a}_{\mu} = \beta^{a}v_{\mu} $?
• RQ3 What is the geometric and algebraic structure of the connection that yields a two-form field, and how does it transform under gauge transformations?
• RQ4 How is the field strength $ {\cal F}_{\mu\nu} $ derived from the covariant derivative, and does it reproduce the standard Kalb-Ramond field strength $ H_{\mu\nu\rho} $?
• RQ5 Can this construction be extended to non-Abelian theories and coupled to fermionic matter in a gauge-invariant way?

Key findings

• The Kalb-Ramond field $ b_{\mu\nu} $ emerges as the antisymmetric part of a rank-2 tensor connection $ B_{\mu\nu} $, which is constructed from a tensor product of gauge groups and Dirac matrices.
• The field strength derived from the covariant derivative is $ {\cal F}_{\mu\nu\rho} = 2\partial_{[\mu}B_{\nu]\rho} $, which reduces to the standard Kalb-Ramond field strength $ H_{\mu\nu\rho} $ when fully antisymmetrized.
• The gauge transformation law for the connection is $ B'_{\mu} - B_{\mu} = (\partial_{\mu}\beta_{\nu})\Gamma^{\nu} $, confirming that $ B_{\mu} $ transforms as a two-form under gauge symmetry.
• The construction removes the need for the separability condition $ \beta^{a}_{\mu} = \beta^{a}v_{\mu} $, allowing the full gauge parameter to be non-separable and more general.
• The framework is manifestly Lorentz-covariant, with matrices $ \Gamma^{\mu}_{(i)} $ transforming properly under Lorentz transformations, ensuring spacetime consistency.
• The method provides a geometric foundation for extending the theory to non-Abelian gauge groups and coupling to fermions via a consistent gauge-covariant derivative.