Abstract
We introduce prepotentials for fermionic higher-spin gauge fields in four space-time dimensions, generalizing earlier work on bosonic fields. To that end, we first develop tools for handling conformal fermionic higher-spin gauge fields in three dimensions. This is necessary because the prepotentials turn out to be three-dimensional fields that enjoy both “higher-spin diffeomorphism” and “higher-spin Weyl” gauge symmetries. We discuss a number of the key properties of the relevant Cotton tensors. The reformulation of the equations of motion as “twisted self-duality conditions” is then exhibited. We show next how the Hamiltonian constraints can be explicitly solved in terms of appropriate prepotentials and show that the action takes then the same remarkable form for all spins.
Highlights
The difficulty with dimension three is that conformal symmetry is not controlled by the Weyl tensor, which identically vanishes, but by the Cotton tensor, which involves higher derivatives of the fields
The Einstein tensor and its derivatives provide a complete set of higher-spin diffeomorphism invariant functions, but little can be said about higher-spin Weyl symmetry without introducing the Cotton tensor
We have developed the conformal geometry of higher-spin fermionic fields in three dimensions
Summary
The Riemann tensor, or equivalently the Einstein tensor, controls higher-spin diffeomorphisms. By this we mean that any function that is higher-spin diffeomorphism invariant can be written as a function of the Riemann (or equivalently Einstein) tensor and its derivatives. The Riemann tensor lacks higher-spin conformal invariance, which is an important property needed for the resolution of the Hamiltonian constraints. For this reason the Cotton tensor must be introduced and its important properties established
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