BRST construction of interacting gauge theories of higher spin fields

The problem of constructing consistent interacting theories of an infinite tower of high-spin fields is formulated in terms of a field φ(X) defined on a finite- or infinite-dimensional extended space-time manifold , generalizing the concept of a string-field φ[x(σ)]. Using the techniques of BRST cohomology we first analyze the conditions under which a system of compatible wave equations for φ defines a gauge-invariant free theory without ghosts in its physical sector. The necessary and sufficient conditions for the existence of a gauge-invariant cubic interaction are then shown to take the concise form of a graded Lie algebra, on which the BRST charge acts as an exterior derivative. Witten's string-field theory is a particular realization of this algebra. The importance of a search for other realizations, possibly operating on a finite-dimensional manifold and/or leading to an interacting theory of massless gauge fields of any spin, is stressed and commented on.