[Paper Review] Many faces of Born-Infeld theory

Sergei V. Ketov|ArXiv.org|Aug 25, 2001

Black Holes and Theoretical Physics1 references22 citations

TL;DR

This paper reviews the Born-Infeld (BI) theory as a non-linear electrodynamics that tames singularities, ensures causal propagation, and exhibits electric-magnetic duality. It extends BI to $N=1$ and $N=2$ supersymmetric theories and constructs a non-linear supergravity action—Born-Infeld-Weyl (BIW)—that tames spacetime curvature, yielding a finite upper bound on the Ricci scalar and a gravitational analogue of the Euler-Heisenberg term via the Bel-Robinson tensor squared.

ABSTRACT

Born-Infeld theory is the non-linear generalization of Maxwell electrodynamics. It naturally arises as the low-energy effective action of open strings, and it is also part of the world-volume effective action of D-branes. The N=1 and N=2 supersymmetric generalizations of the Born-Infeld action are closely related to partial spontaneous breaking of rigid extended supersymmetry. We review some remarkable features of the Born-Infeld action and outline its supersymmetric generalizations in four dimensions. The non-abelian N=1 supersymmetric extension of the Born-Infeld theory and its N=1 supergravitational avatars are given in superspace.

Motivation & Objective

• To review the foundational properties of Born-Infeld electrodynamics, including its regularization of Coulomb self-energy and shockwave-free propagation.
• To explore its supersymmetric extensions in $N=1$ and $N=2$ supersymmetry, particularly in relation to partial spontaneous breaking of extended supersymmetry.
• To construct a non-linear $N=1$ supergravity action—Born-Infeld-Weyl (BIW)—that generalizes the BI Lagrangian to gravity, ensuring finiteness of curvature.
• To establish the gravitational analogue of the gauge Euler-Heisenberg term through the Bel-Robinson tensor squared in the BIW action.

Proposed method

• Formulates the Born-Infeld Lagrangian via the determinant of the metric deformation: ${ m L}_{ m BI} = rac{1}{b^2} ig(1 - ig| ext{det}( ilde{ heta}^{ ho au} + bF_{ ho au}) ig|^{1/2} ig)$.
• Uses superspace techniques to construct $N=1$ and $N=2$ supersymmetric extensions of the BI action, ensuring off-shell closure and duality symmetry.
• Derives the BIW supergravity action via a non-linear superfield constraint: ${ m f F} = rac{1}{2} { m f F} (ar{ m D}^2 - 4{ m R}) ar{ m F} + W^2$, with chiral density ${ m f E}$.
• Introduces a leading-order correction to the Weyl supergravity action via the Bel-Robinson tensor squared: $T_{mnpq}^2 = R_{mspt}R_n{}^s{}_q{}^t + ilde{R}_{mspt} ilde{R}_n{}^s{}_q{}^t$.
• Analyzes the bosonic component of the BIW action, showing it yields a curvature-squared term that bounds the Ricci scalar from above.
• Derives the effective action after solving for auxiliary fields, resulting in a BI-type action with a finite upper bound on $R$.

Experimental results

Research questions

• RQ1 How does the Born-Infeld theory resolve the Coulomb self-energy singularity of point charges while preserving causality and duality?
• RQ2 What are the supersymmetric generalizations of Born-Infeld theory in $N=1$ and $N=2$ supersymmetry, and how do they relate to partial spontaneous breaking of extended supersymmetry?
• RQ3 Can a non-linear supergravity action be constructed that generalizes the Born-Infeld Lagrangian to gravity, ensuring finiteness of spacetime curvature?
• RQ4 What is the gravitational analogue of the gauge Euler-Heisenberg term in the context of supergravity, and how is it realized via the Bel-Robinson tensor?
• RQ5 How does the effective action of the BIW supergravity model yield a finite upper bound on the Ricci scalar $R$?

Key findings

• The Born-Infeld theory eliminates the Coulomb self-energy singularity: the electric field of a point charge is bounded as $E o Q / ig( r^2 + Q^2 ig)^{1/2} $, ensuring finite total energy.
• The theory satisfies the dominant energy condition and avoids shock waves due to phase-independent phase speed, confirming causal propagation.
• The $N=1$ supersymmetric extension of BI theory is constructed in superspace using a non-linear superfield constraint involving the chiral superfield $W^2$ and the supercurvature ${ m f R}$.
• The Born-Infeld-Weyl (BIW) supergravity action is derived as $S_{ m BIW} = ext{Re} ig[ extstylerac{1}{2} ig( extstylerac{1}{2} ext{tr} ig( { m f F} (ar{ m D}^2 - 4{ m R}) ar{ m F} ig) + W^2 ig) ig] $, with a non-linear constraint on the chiral superfield ${ m f F}$.
• The leading correction to the Weyl supergravity action is proportional to the square of the Bel-Robinson tensor, $T_{mnpq}^2$, which serves as the gravitational analogue of the gauge Euler-Heisenberg term.
• After solving the auxiliary field equation, the effective bosonic action yields a finite upper bound on the Ricci scalar: $R ig( ext{max} ig) = rac{1}{3} ig( rac{3}{2 ilde{ heta}} ig)^2 $, with $ ilde{ heta} = rac{1}{ ilde{ heta}} = rac{3}{2 ilde{ heta}} $, ensuring curvature finiteness.