Three-dimensional conformal geometry and prepotentials for four-dimensional fermionic higher-spin fields
Marc Henneaux (Université Libre de Bruxelles and International Solvay Institutes, Campus Plaine — CP 231, Bruxelles, B-1050, Belgium); Victor Lekeu (Université Libre de Bruxelles and International Solvay Institutes, Campus Plaine — CP 231, Bruxelles, B-1050, Belgium, The Blackett Laboratory, Imperial College London, Prince Consort Road, London, SW7 2AZ, U.K.); Amaury Leonard (Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, Potsdam, DE-14476, Germany, Université Libre de Bruxelles and International Solvay Institutes, Campus Plaine — CP 231, Bruxelles, B-1050, Belgium); Javier Matulich (Université Libre de Bruxelles and International Solvay Institutes, Campus Plaine — CP 231, Bruxelles, B-1050, Belgium); Stefan Prohazka (Université Libre de Bruxelles and International Solvay Institutes, Campus Plaine — CP 231, Bruxelles, B-1050, Belgium)
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.
