Abstract
Over the past two decades, electric-magnetic duality has made significant progress in linearized gravity and higher spin gauge fields in arbitrary dimensions. By analogy with Maxwell theory, the Dirac quantization condition has been generalized to both the conserved electric-type and magnetic-type sources associated with gravitational fields and higher spin fields. The linearized Einstein equations in $D$ dimensions, which are expressed in terms of the Pauli-Fierz field of the \textit{graviton} described by a 2nd-rank symmetric tensor, can be dual to the linearized field equations of the \textit{dual graviton} described by a Young symmetry $(D-3, 1)$ tensor. Hence, the dual formulations of linearized gravity are written by a 2nd-rank symmetric tensor describing the Pauli-Fierz field of the dual graviton in $D=4$, while we have the Curtright field with Young symmetry type $(2,1)$ in $D=5$. The equations of motion of spin-$s$ fields ($s>2$) described by the generalized Fronsdal action can also be dualized to the equations of motion of dual spin-$s$ fields. In this review, we focus on dual formulations of gravity and higher spin fields in the linearized theory, and study their SO(2) electric-magnetic duality invariance, twisted self-duality conditions, harmonic conditions for wave solutions, and their configurations with electric-type and magnetic-type sources. Furthermore, we briefly discuss the latest developments in their interacting theories.
Highlights
Electric-magnetic duality basically evolved from the invariance of Maxwell’s equations [1], which led to the hypothesis about magnetic monopoles in electromagnetism [2], and the introduction of magnetic-type sources to gauge theories [3,4,5,6,7]
From Equations (147, 148), the wave equation of the Pauli-Fierz field hμν in empty space, ∂ρ ∂ρ hμν = 0, corresponds to ∂ρ ∂ρ Eμν = 0 = ∂ρ ∂ρ Bμν in the linearized theory, which are equivalent to D2Eμν − Eμν ′′ = 0 = D2Bμν − Bμν ′′ in the 1 + 3 covariant formalism [60, 66]
Consistency of the interacting theory for a free field can traditionally be determined from coupling deformations of the gauge transformations of the free theory
Summary
Electric-magnetic duality basically evolved from the invariance of Maxwell’s equations [1], which led to the hypothesis about magnetic monopoles in electromagnetism [2], and the introduction of magnetic-type sources to gauge theories [3,4,5,6,7]. The linearized formulations of gravity describe the graviton using a 2-rank symmetric tensor hμν, which can be dualized to a gauge field hμ1···μD−3ν with mixed symmetry (D − 3, 1), the so-called dual graviton. The twisted self-duality conditions, which were previously discussed in dual formulations of linearized gravity [43, 108], were illustrated in higher spin gauge fields, which maintain an SO(2) electric-magnetic invariance in D = 4 [49]. In what follows we evaluate general aspects of electricmagnetic duality in gravity and higher spin field theories (s ≥ 2) in arbitrary dimensions (D ≥ 4), and advocate their formulations in linearized theories, their couplings with electricand magnetic-type sources, and their harmonic conditions for wave propagation.
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