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An action for higher spin gauge theory in four dimensions

An action principle is presented for Vasiliev's Bosonic higher spin gauge theory in four spacetime dimensions. The action is of the form of a broken topological field theory, and arises by an extension of the MacDowell-Mansouri formulation of general relativity. In the latter theory the local degrees of freedom of general relativity arise by breaking the gauge invariance of a topological theory from $sp(4)$ to the Lorentz algebra. In Vasiliev's theory the infinite number of degrees of freedom with higher spins similarly arise by the breaking of a topological theory with an infinite dimensional gauge symmetry extending $sp(4)$ to the Lorentz algebra. The Hamiltonian formulation of Vasilev's theory is then derived from our action, and it is shown that the Hamiltonian is a linear combination of constraints, as expected for a diffeomorphism invariant theory. The constraint algebra is computed and found to be first class.