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Inconsistency of interacting, multi-graviton theories

Nicolas Boulanger, Thibault Damour, Leonardo Gualtieri, Marc Henneaux

TL;DR

The work addresses whether multiple massless spin-2 fields can be consistently coupled in a ghost-free, local theory with at most two derivatives. Using BRST deformation theory and characteristic cohomology, the authors show that any consistent deformation must yield a direct sum of diffeomorphism algebras and, at two derivatives, a sum of independent Einstein–Hilbert actions; cross-interactions between different gravitons are therefore forbidden. Matter couplings do not remedy this prohibition, and even in the infinite-field limit the result generalizes to a parallel-set of graviton sectors. Only in cases with a non-positive internal metric or higher-derivative terms could cross-interactions appear, but these typically introduce ghosts or pathological features. The findings reinforce the special status of the Einstein theory as the unique ghost-free, multi-graviton structure under the stated assumptions, and illuminate the precise cohomological obstructions to more elaborate couplings.

Abstract

We investigate, in any spacetime dimension >=3, the problem of consistent couplings for a finite collection of massless, spin-2 fields described, in the free limit, by a sum of Pauli-Fierz actions. We show that there is no consistent (ghost-free) coupling, with at most two derivatives of the fields, that can mix the various "gravitons". In other words, there are no Yang-Mills-like spin-2 theories. The only possible deformations are given by a sum of individual Einstein-Hilbert actions. The impossibility of cross-couplings subsists in the presence of scalar matter. Our approach is based on the BRST-based deformation point of view and uses results on the so-called "characteristic cohomology" for massless spin-2 fields which are explained in detail.

Inconsistency of interacting, multi-graviton theories

TL;DR

The work addresses whether multiple massless spin-2 fields can be consistently coupled in a ghost-free, local theory with at most two derivatives. Using BRST deformation theory and characteristic cohomology, the authors show that any consistent deformation must yield a direct sum of diffeomorphism algebras and, at two derivatives, a sum of independent Einstein–Hilbert actions; cross-interactions between different gravitons are therefore forbidden. Matter couplings do not remedy this prohibition, and even in the infinite-field limit the result generalizes to a parallel-set of graviton sectors. Only in cases with a non-positive internal metric or higher-derivative terms could cross-interactions appear, but these typically introduce ghosts or pathological features. The findings reinforce the special status of the Einstein theory as the unique ghost-free, multi-graviton structure under the stated assumptions, and illuminate the precise cohomological obstructions to more elaborate couplings.

Abstract

We investigate, in any spacetime dimension >=3, the problem of consistent couplings for a finite collection of massless, spin-2 fields described, in the free limit, by a sum of Pauli-Fierz actions. We show that there is no consistent (ghost-free) coupling, with at most two derivatives of the fields, that can mix the various "gravitons". In other words, there are no Yang-Mills-like spin-2 theories. The only possible deformations are given by a sum of individual Einstein-Hilbert actions. The impossibility of cross-couplings subsists in the presence of scalar matter. Our approach is based on the BRST-based deformation point of view and uses results on the so-called "characteristic cohomology" for massless spin-2 fields which are explained in detail.

Paper Structure

This paper contains 30 sections, 9 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Cohomological reformulation
  3. Gauge symmetries and master equation
  4. Perturbation of the master equation
  5. BRST transformation, first order deformations, obstructions
  6. Solution of the master equation for a collection of free, spin-2, massless fields
  7. Cohomology of $\gamma$
  8. Characteristic cohomology -- cohomology of $\delta$ modulo $d$
  9. Characteristic cohomology
  10. Cohomology of $\delta$ modulo $d$
  11. Invariant cohomology of $\delta$ modulo $d$
  12. Construction of the general gauge theory of interacting gravitons by means of cohomological techniques
  13. Determination of $a_2$
  14. Determination of $a_1$
  15. Determination of $a_0$
  16. ...and 15 more sections

Key Result

Theorem 1.1

Under the assumptions of: locality, Poincaré invariance, Eq.(startingpoint) as free field limit and at most two derivatives in the Lagrangian, the only consistent deformation of Eq.(startingpoint) involving a collection of spin-2 fields is (modulo field redefinitions) a sum of independent Einstein-H where $R^a$ is the scalar curvature of $g^a_{\mu \nu}$, $g^a$ its determinant, $\kappa^a \geq 0$ a

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.1
  • Theorem A.1
  • Lemma A.1
  • Lemma A.2