Gauge Theory of Relativistic Membranes

TL;DR

This paper demonstrates an equivalence between relativistic membranes and Kalb-Ramond gauge fields, providing new formulations and extending the correspondence to p-branes coupled with gravity in arbitrary dimensions.

Contribution

It introduces a novel first-order Lagrangian for membranes and establishes a general equivalence with gauge fields in various spacetime dimensions.

Findings

Membranes can be represented by Kalb-Ramond gauge fields satisfying Maxwell-type equations.

A Hamilton-Jacobi formulation for membranes is developed using differential forms.

The equivalence extends to p-branes coupled to gravity in arbitrary dimensions.

Abstract

In this paper we show that a relativistic membrane admits an equivalent representation in terms of the Kalb-Ramond gauge field $F_{\mu\nu\rho}=\partial_{\,[\,\mu}B_{\nu\rho]}$ encountered in string theory. By `` equivalence '' we mean the following: if $x=X(\xi)$ is a solution of the classical equations of motion derived from the Dirac-Nambu-Goto action, then it is always possible to find a differential form of {\it rank three}, satisfying Maxwell-type equations. The converse proposition is also true. In the first part of the paper, we show that a relativistic membrane, regarded as a mechanical system, admits a Hamilton-Jacobi formulation in which the H-J function describing a family of classical membrane histories is given by $\displaystyle{F=dB=dS^1\wedge dS^2\wedge dS^3}$. In the second part of the paper, we introduce a {\it new} lagrangian of the Kalb-Ramond type which provides a {\it first order} formulation for both open and closed membranes. Finally, for completeness, we show that such a correspondence can be established in the very general case of a p-brane coupled to gravity in a spacetime of arbitrary dimensionality.