Uniqueness of the Freedman-Townsend Interaction Vertex For Two-Form Gauge Fields

Abstract

Let $B_{μν}^a$ ($a=1,...,N$) be a system of $N$ free two-form gauge fields, with field strengths $H_{μνρ}^a = 3 \partial _{[μ}B_{νρ]}^a$ and free action $S_0$ equal to $(-1/12)\int d^nx\ g_{ab}H_{μνρ}^aH^{bμνρ}$ ($n\geq 4$). 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 $B_{μν}^a\rightarrow B_{μν}^a+\partial _{[μ}Λ_{ν]}$ 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.