No Self-Interaction for Two-Column Massless Fields

TL;DR

This work investigates whether massless mixed-symmetry gauge fields with two-column Young diagrams $[p,q]$ (with $p>q$) in flat space can admit consistent self-interactions. Using a BRST-cohomology deformation framework, it classifies potential first-order vertices via $H^{D,0}(s|d)$ and derives the invariant cohomologies and Poincaré lemmas that organize possible deformations. The main result is a no-go: under locality and translation invariance, there is no local, smooth deformation with at most two derivatives that deforms the gauge algebra for $p\neq q$, and deformations that modify gauge transformations are generically excluded; even relaxing derivative counting yields no viable interactions in most cases. When a special arithmetic condition $p+2=(s+1)(q+1)$ holds, certain higher-derivative candidates can appear but fail to be Lorentz-invariant and derivative-bounded within the required class, so they do not provide viable Yang-Mills– or Einstein-type interactions for these fields. Together with prior results for $p=q$, these findings substantially constrain the existence of interacting theories for two-column mixed-symmetry fields in flat space and point to fundamental obstructions to generalizing standard gauge interactions to these exotic higher-spin sectors.

Abstract

We investigate the problem of introducing consistent self-couplings in free theories for mixed tensor gauge fields whose symmetry properties are characterized by Young diagrams made of two columns of arbitrary (but different) lengths. We prove that, in flat space, these theories admit no local, Poincaré-invariant, smooth, self-interacting deformation with at most two derivatives in the Lagrangian. Relaxing the derivative and Lorentz-invariance assumptions, there still is no deformation that modifies the gauge algebra, and in most cases no deformation that alters the gauge transformations.Our approach is based on a BRST-cohomology deformation procedure.

Figures (1)

• Figure 1: Antighost-zero BRST spectrum of $[p,q]-$type gauge field.

Theorems & Definitions (16)

• Theorem 4.1
• Lemma 5.1: Invariant Poincaré lemma in form degree $k<p+1$
• Theorem 6.1
• Theorem 6.2
• Theorem 6.3
• Theorem 6.4
• Theorem 6.5
• Theorem 7.1
• Theorem 7.3
• Lemma 7.1