An action for higher spin gauge theory in four dimensions

TL;DR

We address the lack of a covariant action for 4D bosonic higher-spin gauge theory by formulating Vasiliev’s equations as the Euler–Lagrange equations of a broken topological action. The construction extends the MacDowell–Mansouri program using the HS algebra $hs(4)$, a $Z$-extension, and the unfolded/free differential-algebra framework, yielding an action with Lagrange multipliers whose variations reproduce Vasiliev’s equations. A full Hamiltonian analysis confirms the constraints are first-class and that the Hamiltonian is a linear combination of these constraints, with the Gauss constraint generating gauge transformations and, on the constraint surface, spatial diffeomorphisms. This work thus unifies higher-spin interactions with a broken-topological-field-theory perspective and provides a practical route toward quantization and holographic interpretations in AdS$_4$/CFT$_3$.

Abstract

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.