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Parity violating vertices for spin-3 gauge fields

Nicolas Boulanger, Sandrine Cnockaert, Serge Leclercq

Abstract

The problem of constructing consistent parity-violating interactions for spin-3 gauge fields is considered in Minkowski space. Under the assumptions of locality, Poincaré invariance and parity non-invariance, we classify all the nontrivial perturbative deformations of the abelian gauge algebra. In space-time dimensions $n=3$ and $n=5$, deformations of the free theory are obtained which make the gauge algebra non-abelian and give rise to nontrivial cubic vertices in the Lagrangian, at first order in the deformation parameter $g$. At second order in $g$, consistency conditions are obtained which the five-dimensional vertex obeys, but which rule out the $n=3$ candidate. Moreover, in the five-dimensional first order deformation case, the gauge transformations are modified by a new term which involves the second de Wit--Freedman connection in a simple and suggestive way.

Parity violating vertices for spin-3 gauge fields

Abstract

The problem of constructing consistent parity-violating interactions for spin-3 gauge fields is considered in Minkowski space. Under the assumptions of locality, Poincaré invariance and parity non-invariance, we classify all the nontrivial perturbative deformations of the abelian gauge algebra. In space-time dimensions and , deformations of the free theory are obtained which make the gauge algebra non-abelian and give rise to nontrivial cubic vertices in the Lagrangian, at first order in the deformation parameter . At second order in , consistency conditions are obtained which the five-dimensional vertex obeys, but which rule out the candidate. Moreover, in the five-dimensional first order deformation case, the gauge transformations are modified by a new term which involves the second de Wit--Freedman connection in a simple and suggestive way.

Paper Structure

This paper contains 22 sections, 5 theorems, 47 equations, 1 table.

Table of Contents

  1. Introduction
  2. Free theory
  3. BRST formulation
  4. Basic assumptions
  5. BRST spectrum and differential
  6. BRST deformation
  7. Cohomology of $\gamma$
  8. Cohomological group $H^n_2(\delta| d)$ and $H^{n}_k(\delta| d,H(\gamma))$, $k\geqslant 2$
  9. Parity-breaking deformations
  10. Deformations of the gauge algebra
  11. Deformation in 3 dimensions
  12. First-order deformation
  13. Second-order deformation
  14. Deformation in 5 dimensions
  15. First-order deformation
  16. ...and 7 more sections

Key Result

Proposition 1

The cohomology of $\gamma$ is isomorphic to the space of functions depending on Thus, identifying with zero any $\gamma$-exact term in $H(\gamma)$, we have $\gamma f=0$ if and only if $f= f\left([\Phi^{*i}],[K^a_{\alpha\mu|\beta\nu|\gamma\rho}],\{F^a_{\mu\nu\rho}\}, C_{\mu\nu}^a, \widehat{T}^a_{\alpha\mu\vert\nu}, \widehat{U}^a_{\alpha\mu\vert\beta\nu} \right)$ where $\{F^a_{\m

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Theorem 1