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We show that the equations of motion for (free) integer higher spin gaugefields can be formulated as twisted self-duality conditions on the higher spincurvatures of the spin-$s$ field and its dual. We focus on the case of fourspacetime dimensions, but formulate our results in a manner applicable tohigher spacetime dimensions. The twisted self-duality conditions are redundantand we exhibit a non-redundant subset of conditions, which have the remarkableproperty to involve only first-order derivatives with respect to time. Thisnon-redundant subset equates the electric field of the spin-$s$ field (which wedefine) to the magnetic field of its dual (which we also define), and viceversa. The non-redundant subset of twisted self-duality conditions involve thepurely spatial components of the spin-$s$ field and its dual, and also thecomponents of the fields with one zero index. One can get rid of these gaugecomponents by taking the curl of the equations, which does not change theirphysical content. In this form, the twisted self-duality conditions can bederived from a variational principle that involves prepotentials, which are thehigher spin generalizations of the prepotentials previously found in the spins2 and 3 cases. The prepotentials have again the intriguing feature ofpossessing both higher spin diffeomorphism invariance and higher spin conformalgeometry. The tools introduced in an earlier paper for handling higher spinconformal geometry turn out to be crucial for streamlining the analysis. Infour spacetime dimensions where the electric and magnetic fields are tensorfields 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 theequations of motion). Remarks on the extension to higher dimensions are given.

Twisted self-duality for higher spin gauge fields and prepotentials