Interactions of a massless tensor field with the mixed symmetryof the Riemann tensor. No-go results

Abstract

Non-trivial, consistent interactions of a free, massless tensor field t µν|αβ with the mixed symmetry of the Riemann tensor are studied in the following cases: self-couplings, cross-interactions with a Pauli-Fierz field and cross-couplings with purely matter theories. The main results, obtained from BRST cohomological techniques under the assumptions on smoothness, locality, Lorentz covariance and Poincaré invariance of the deformations, combined with the requirement that the interacting Lagrangian is at most second-order derivative, can be synthesized into: no consistent self-couplings exist, but a cosmologicallike term; no cross-interactions with the Pauli-Fierz field can be added; no non-trivial consistent cross-couplings with the matter theories such that the matter fields gain gauge transformations are allowed.

COHOMOLOGICAL BRST ASPECTS OF THE MASSLESS TENSOR FIELD WITH THE MIXED SYMMETRY (k,k)

“…The construction of the Lagrangian action for such a tensor field relies on the general principle of gauge invariance, combined with the requirements of locality, Lorentz covariance, Poincaré invariance, zero mass and the natural assumptions that the field equations are linear in the field, second-order derivative and do not break the PT invariance. In view of all these, a natural point to start with is to stipulate the (infinitesimal) gauge invariance of the action such that to recover the linearized limit of diffeomorphisms for k = 1 and the gauge symmetry [5,6] of the free, massless tensor field with the mixed symmetry of the Riemann tensor for k = 2. The simplest way to achieve this is to ask that the Lagrangian action S L t µ 1 ···µ k |ν 1 ···ν k is invariant under the (infinitesimal) gauge transformations…”

“…being understood that we maintain the conventions (6). By direct computation, from (30) we get that the various traces of the curvature tensor have the expressions…”

“…By direct computation, from (30) we get that the various traces of the curvature tensor have the expressions…”

“…To put it otherwise, if this tensor satisfies the Bianchi II identity ∂ [µ 1 F µ 2 ···µ k+2 ]|ν 1 ···ν k+1 ≡ 0, then there exists an element t µ 1 ···µ k |ν 1 ···ν k with the mixed symmetry (1), with the help of which F µ 1 ···µ k+1 |ν 1 ···ν k+1 can precisely be written like in (30).…”

“…This type of models became of special interest lately due to the many desirable featured exhibited, like the dual formulation of field theories of spin two or higher [7,8,9,10,11,12], the impossibility of consistent cross-interactions in the dual formulation of linearized gravity [13] or a Lagrangian first-order approach [14,15] to some classes of massless or partially massive mixed symmetry-type tensor gauge fields, suggestively resembling to the tetrad formalism of General Relativity. A basic problem involving mixed symmetrytype tensor fields is the approach to their local BRST cohomology, since it is helpful at solving many Lagrangian and Hamiltonian aspects, like, for instance the determination of their consistent interactions [16] with higher-spin gauge theories [6,18,19,20,21,22,30,31]. The present paper proposes the investigation of the basic cohomological ingredients involved in the structure of the co-cycles from the local BRST cohomology for a free, massless tensor gauge field t µ 1 ···µ k |ν 1 ···ν k that transforms in an irreducible representation of GL (D, R), corresponding to a rectangular, two-column Young diagram with k > 2 rows.…”