Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Strong obstruction of the Berends-Burgers-van Dam spin-3 vertex

Xavier Bekaert, Nicolas Boulanger, Serge Leclercq

TL;DR

The paper proves a strong obstruction to the Berends–Burgers–van Dam spin-3 vertex in flat spacetime, showing that higher-spin extensions cannot cure the spin-3 obstruction at second order. Using the antifield (BV) framework, it derives general constraints on cubic gauge-algebra deformations (a2) for symmetric tensor fields and analyzes derivative counting via γ-cohomology, ensuring only specific a2 structures can contribute. The BBvD obstruction arises from non-γ-exact terms in (a2,a2), and detailed examination of the 3-3-4 and 3-3-5 cases demonstrates that no compatible a2 can cancel it, regardless of introducing higher spins. The results suggest that consistent nonabelian higher-spin interactions in flat spacetime are unlikely beyond the cubic level, with AdS (Fradkin–Vasiliev) construction remaining a viable route due to its altered derivative grading.

Abstract

In the eighties, Berends, Burgers and van Dam (BBvD) found a nonabelian cubic vertex for self-interacting massless fields of spin three in flat spacetime. However, they also found that this deformation is inconsistent at higher order for any multiplet of spin-three fields. For arbitrary symmetric gauge fields, we severely constrain the possible nonabelian deformations of the gauge algebra and, using these results, prove that the BBvD obstruction cannot be cured by any means, even by introducing fields of spin higher (or lower) than three.

Strong obstruction of the Berends-Burgers-van Dam spin-3 vertex

TL;DR

The paper proves a strong obstruction to the Berends–Burgers–van Dam spin-3 vertex in flat spacetime, showing that higher-spin extensions cannot cure the spin-3 obstruction at second order. Using the antifield (BV) framework, it derives general constraints on cubic gauge-algebra deformations (a2) for symmetric tensor fields and analyzes derivative counting via γ-cohomology, ensuring only specific a2 structures can contribute. The BBvD obstruction arises from non-γ-exact terms in (a2,a2), and detailed examination of the 3-3-4 and 3-3-5 cases demonstrates that no compatible a2 can cancel it, regardless of introducing higher spins. The results suggest that consistent nonabelian higher-spin interactions in flat spacetime are unlikely beyond the cubic level, with AdS (Fradkin–Vasiliev) construction remaining a viable route due to its altered derivative grading.

Abstract

In the eighties, Berends, Burgers and van Dam (BBvD) found a nonabelian cubic vertex for self-interacting massless fields of spin three in flat spacetime. However, they also found that this deformation is inconsistent at higher order for any multiplet of spin-three fields. For arbitrary symmetric gauge fields, we severely constrain the possible nonabelian deformations of the gauge algebra and, using these results, prove that the BBvD obstruction cannot be cured by any means, even by introducing fields of spin higher (or lower) than three.

Paper Structure

This paper contains 16 sections, 1 theorem, 33 equations.

Table of Contents

  1. Introduction
  2. Antifield formulation and consistent deformations
  3. The Fronsdal Lagrangian
  4. Antifield formulation
  5. Cohomology of $\gamma$
  6. Consistent deformations
  7. Cubic vertices
  8. General results on the gauge algebra deformations
  9. Product of ghost tensors
  10. Bound on the difference of the number of derivatives
  11. Conditions on the total number of derivatives
  12. A general theorem and a particular candidate
  13. Proof of the strong obstruction on the BBvD vertex
  14. Study of $a_2$ in the $3-3-4$ case
  15. Study of $a_2$ in the $3-3-5$ case
  16. ...and 1 more sections

Key Result

Theorem 1

Given a cubic configuration of fields with spins $s\leqslant s'\leqslant s"$, the possible Poincaré invariants $a_2=C^* U^{(i)}U^{(j)}\,d^nx$ are contractions of an undifferentiated antifield number-$2$ antighost and of two ghost tensors, involving $i$ and $j$ derivatives. The spins and the numbers

Theorems & Definitions (1)

  • Theorem