Restrictions for $n$-Point Vertices in Higher-Spin Theories

TL;DR

This paper classifies independent interaction vertices for bosonic higher-spin fields in various dimensions, revealing how dimensionality and Schouten identities influence the structure and restrictions of these vertices.

Contribution

It provides a comprehensive classification of n-point vertices in higher-spin theories across different dimensions, highlighting the role of Schouten identities and gauge invariance.

Findings

Vertices in high dimensions are unaffected by Schouten identities.

Lower dimensions impose strong restrictions on vertices due to Schouten identities.

Vertices are expressed in gauge-invariant terms, indicating no gauge deformation occurs.

Abstract

We give a simple classification of the independent $n$-point interaction vertices for bosonic higher-spin gauge fields in $d$-dimensional Minkowski space-times. We first give a characterisation of such vertices for large dimensions, $d \geq 2n - 1$, where one does not have to consider Schouten identities due to over-antisymmetrisation of space-time indices. When the dimension is lowered, such identities have to be considered, but their appearance only leads to equivalences of large-$d$ vertices and does not lead to new types of vertices. We consider the case of low dimensions, $d<n$, in detail, where the large number of Schouten identities leads to strong restrictions on independent vertices. We also comment on the generalisation of our results to the intermediate case $n \leq d \leq 2n - 2$. In all cases, the independent vertices are expressed in terms of elementary manifestly gauge-invariant quantities, suggesting that no deformations of the gauge transformations are induced.