Uniqueness of the Freedman-Townsend interaction vertex for two-form gauge fields

Let Bμνa (a = 1,…,N) be a system of N free two-form gauge fields, with field strengths Hμνϱa = 3∂[μBνϱa] and free action S0 equal to (−112)∫dnx gabHaμvpHbμvp (n ≥ 4). It is shown that in n > 4 dimensions, the only consistent local interactions that can be added to the free action are given by functions of the field strength components and their derivatives (and the Chern-Simons forms in 5 mod 3 dimensions). These interactions do not modify the gauge invariance Bμνa → Bμνa + ∂[μΛν] of the free theory. By contrast, there exist in n = 4 dimensions consistent interactions that deform the gauge symmetry of the free theory in a non trivial way. These consistent interactions are uniquely given by the well-known Freedman-Townsend vertex. The method of proof uses the cohomological techniques developed recently in the Yang-Mills context to establish theorems on the structure of renormalized gauge-invariant operators.