Higher-Spin Fermionic Gauge Fields and Their Electromagnetic Coupling

TL;DR

This work systematically classifies consistency constraints for electromagnetic couplings of massless fermionic higher-spin fields in flat space using BRST-BV cohomology. It shows minimal coupling is forbidden for spins $s\ge \tfrac{3}{2}$ and constrains cubic vertices to derivative counts of $2n-1$, $2n$, or $2n+1$ for spin $s=n+\tfrac{1}{2}$. Among these, the $(2n-1)$-derivative vertex non-trivially deforms the gauge algebra, while the others are Abelian; many Abelian cases preserve gauge symmetries and serve as Lagrangian deformations only. Second-order consistency obstructions prevent non-abelian vertices from surviving in a local theory without extra dynamical higher-spin fields (e.g., gravity), aligning with no-go expectations yet providing explicit covariant constructions. The results agree with Metsaev’s and Sagnotti–Taronna’s findings, and establish off-shell equivalence with their TT-gauge on-shell forms, while highlighting the role of curvature terms (Chern–Simons and Born–Infeld–type structures) in higher-derivative Abelian vertices. These insights illuminate the landscape of higher-spin interactions and their potential extensions to AdS and non-local frameworks.

Abstract

We study the electromagnetic coupling of massless higher-spin fermions in flat space. Under the assumptions of locality and Poincare invariance, we employ the BRST-BV cohomological methods to construct consistent parity-preserving off-shell cubic 1-s-s vertices. Consistency and non-triviality of the deformations not only rule out minimal coupling, but also restrict the possible number of derivatives. Our findings are in complete agreement with, but derived in a manner independent from, the light-cone-formulation results of Metsaev and the string-theory-inspired results of Sagnotti-Taronna. We prove that any gauge-algebra-preserving vertex cannot deform the gauge transformations. We also show that in a local theory, without additional dynamical higher-spin gauge fields, the non-abelian vertices are eliminated by the lack of consistent second-order deformations.