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Spin three gauge theory revisited

Xavier Bekaert, Nicolas Boulanger, Sandrine Cnockaert

Abstract

We study the problem of consistent interactions for spin-3 gauge fields in flat spacetime of arbitrary dimension n>3. Under the sole assumptions of Poincaré and parity invariance, local and perturbative deformation of the free theory, we determine all nontrivial consistent deformations of the abelian gauge algebra and classify the corresponding deformations of the quadratic action, at first order in the deformation parameter. We prove that all such vertices are cubic, contain a total of either three or five derivatives and are uniquely characterized by a rank-three constant tensor (an internal algebra structure constant). The covariant cubic vertex containing three derivatives is the vertex discovered by Berends, Burgers and van Dam, which however leads to inconsistencies at second order in the deformation parameter. In dimensions n>4 and for a completely antisymmetric structure constant tensor, another covariant cubic vertex exists, which contains five derivatives and passes the consistency test where the previous vertex failed.

Spin three gauge theory revisited

Abstract

We study the problem of consistent interactions for spin-3 gauge fields in flat spacetime of arbitrary dimension n>3. Under the sole assumptions of Poincaré and parity invariance, local and perturbative deformation of the free theory, we determine all nontrivial consistent deformations of the abelian gauge algebra and classify the corresponding deformations of the quadratic action, at first order in the deformation parameter. We prove that all such vertices are cubic, contain a total of either three or five derivatives and are uniquely characterized by a rank-three constant tensor (an internal algebra structure constant). The covariant cubic vertex containing three derivatives is the vertex discovered by Berends, Burgers and van Dam, which however leads to inconsistencies at second order in the deformation parameter. In dimensions n>4 and for a completely antisymmetric structure constant tensor, another covariant cubic vertex exists, which contains five derivatives and passes the consistency test where the previous vertex failed.

Paper Structure

This paper contains 23 sections, 12 theorems, 98 equations, 1 table.

Table of Contents

  1. Introduction
  2. Free theory
  3. Deformations of the free theory
  4. Basic assumptions
  5. Main results
  6. BRST settings
  7. BRST spectrum and differential
  8. BRST deformation
  9. Cohomology of $\gamma$
  10. Invariant Poincaré lemma
  11. Cohomology of $\delta$ modulo $d\,$: $H^n_k(\delta \vert\, d)$
  12. Invariant cohomology of $\delta$ modulo $d$: $H_k^{n,inv}(\delta \vert\, d)$
  13. Propagation of the invariance in form degree
  14. Top form degree
  15. Computation of deformations
  16. ...and 8 more sections

Key Result

Theorem 1

Let $h^a_{\mu\nu\rho}$ be a collection of spin-$3$ gauge fields ($a=1,\ldots,N$) described by the local and quadratic action of Fronsdal, in dimension $n>3$. At first order in some smooth deformation parameter, the nontrivial consistent local deformations of the (abelian) gauge algebra that are inva of an anticommutative internal algebra, that may be taken as deformation parameters. Moreover, the

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 3
  • Lemma 2
  • ...and 2 more