Twisted self-duality for higher spin gauge fields and prepotentials

Abstract

We show that the equations of motion for (free) integer higher spin gauge fields can be formulated as twisted self-duality conditions on the higher spin curvatures of the spin-$s$ field and its dual. We focus on the case of four spacetime dimensions, but formulate our results in a manner applicable to higher spacetime dimensions. The twisted self-duality conditions are redundant and we exhibit a non-redundant subset of conditions, which have the remarkable property to involve only first-order derivatives with respect to time. This non-redundant subset equates the electric field of the spin-$s$ field (which we define) to the magnetic field of its dual (which we also define), and vice versa. The non-redundant subset of twisted self-duality conditions involve the purely spatial components of the spin-$s$ field and its dual, and also the components of the fields with one zero index. One can get rid of these gauge components by taking the curl of the equations, which does not change their physical content. In this form, the twisted self-duality conditions can be derived from a variational principle that involves prepotentials, which are the higher spin generalizations of the prepotentials previously found in the spins 2 and 3 cases. The prepotentials have again the intriguing feature of possessing both higher spin diffeomorphism invariance and higher spin conformal geometry. The tools introduced in an earlier paper for handling higher spin conformal geometry turn out to be crucial for streamlining the analysis. In four spacetime dimensions where the electric and magnetic fields are tensor fields of the same type, the twisted self-duality conditions enjoy an $SO(2)$ electric-magnetic invariance. We explicitly show that this symmetry is an "off-shell symmetry" (i.e., a symmetry of the action and not just of the equations of motion). Remarks on the extension to higher dimensions are given.

arXiv commentMinor misprint corrected. Simplification of Hamiltonian added

References in corpus (32)

• E_11 and M Theory
• E10 and a "small tension expansion" of M Theory
• Dualisation of Dualities, I
• Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities
• Tensor gauge fields in arbitrary representations of GL(D,R) : duality & Poincare lemma
• Free geometric equations for higher spins
• Conformal Higher Spin Theory
• Conformal Higher Spin Symmetries of 4d Massless Supermultiplets and $osp(L,2M)$ Invariant Equations in Generalized (Super)Space
• Duality in Gravity and Higher Spin Gauge Fields
• On geometric equations and duality for free higher spins
• Duality in linearized gravity
• Tensor fields of mixed Young symmetry type and N-complexes
• On Higher Derivatives in 3D Gravity and Higher Spin Gauge Theories
• Maxwell-like Lagrangians for higher spins
• E11 in 11D
• Ordinary-derivative formulation of conformal totally symmetric arbitrary spin bosonic fields
• Duality invariance of all free bosonic and fermionic gauge fields
• Gravitational duality near de Sitter space
• The Action for Twisted Self-Duality
• Higher Spin Conformal Geometry in Three Dimensions and Prepotentials for Higher Spin Gauge Fields
• PM = EM: Partially Massless Duality Invariance
• Duality invariance implies Poincare invariance
• A spin-4 analog of 3D massive gravity
• Manifest Duality for Partially Massless Higher Spins
• Gravitational Electric-Magnetic Duality, Gauge Invariance and Twisted Self-Duality
• Twisted Self-Duality for Linearized Gravity in D dimensions
• The non-linear coupled spin 2 - spin 3 Cotton equation in three dimensions
• On the conformal higher spin unfolded equation for a three-dimensional self-interacting scalar field
• Symmetries of higher-spin current interactions in four dimensions
• Exceptional versus superPoincaré algebra as the defining symmetry of maximal supergravity
• Real forms of extended Kac-Moody symmetries and higher spin gauge theories
• Higher-spin charges in Hamiltonian form. I. Bose fields

Cited by in corpus (13)

• Continuous spin gauge field in (A)dS space
• Off-shell massive N=1 supermultiplets in three dimensions
• Chiral Tensors of Mixed Young Symmetry
• Looking for partially-massless gravity
• Dual actions for massless, partially-massless and massive gravitons in (A)dS
• The Schouten tensor as a connection in the unfolding of 3D conformal higher-spin fields
• A note on the double dual graviton
• The Action of the (free) $(4,0)$-theory
• Three-dimensional conformal geometry and prepotentials for four-dimensional fermionic higher-spin fields
• Prepotentials for linearized supergravity
• Gravitational Fields of the Magnetic-type
• Aspects of electric-magnetic dualities in maximal supergravity
• Curvature tensors of higher-spin gauge theories derived from general Lagrangian densities