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

TL;DR

This paper proves that in dimensions higher than four, the only local interactions for free two-form gauge fields are trivial or derivative-based, while in four dimensions, the unique non-trivial interaction is the Freedman-Townsend vertex.

Contribution

It demonstrates the uniqueness of the Freedman-Townsend interaction vertex for two-form gauge fields specifically in four dimensions, contrasting with higher dimensions where interactions are trivial or derivative-based.

Findings

In dimensions >4, only derivative-based or trivial interactions are possible.

In 4 dimensions, the Freedman-Townsend vertex uniquely deforms gauge symmetry.

Interactions in higher dimensions do not alter gauge invariance.

Abstract

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