Gauging the higher-spin-like symmetries by the Moyal product

Abstract

We analyze a novel approach to gauging rigid higher derivative (higher spin) symmetries of free relativistic actions defined on flat spacetime, building on the formalism originally developed by Bonora et al. and Bekaert et al. in their studies of linear coupling of matter fields to an infinite tower of higher spin fields. The off-shell definition is based on fields defined on a 2d-dimensional master space equipped with a symplectic structure, where the infinite dimensional Lie algebra of gauge transformations is given by the Moyal commutator. Using this algebra we construct well-defined weakly non-local actions, both in the gauge and the matter sector, by mimicking the Yang-Mills procedure. The theory allows for a description in terms of an infinite tower of higher spin spacetime fields only on-shell. Interestingly, Euclidean theory allows for such a description also off-shell. Owing to its formal similarity to non-commutative field theories, the formalism allows for the introduction of a covariant potential which plays the role of the generalised vielbein. This covariant formulation uncovers the existence of other phases and shows that the theory can be written in a matrix model form. The symmetries of the theory are analyzed and conserved currents are explicitly constructed. By studying the spin-2 sector we show that the emergent geometry is closely related to teleparallel geometry, in the sense that the induced linear connection is opposite to Weitzenböck’s.

Gauging the Higher-Spin-Like Symmetries by the Moyal Product. II

“…Of course the price one has to pay is that the physical content of the theory, the spectrum in the first place, is far from obvious, making it a legitimate question to what extent MHSYM is directly related to the other previously mentioned approaches. In this paper, we continue the study we started in [54], addressing the above issue in a straightforward way, namely by starting the study of the scattering amplitudes in the context of MHSYM.…”

“…is the MHS torsion (for the origins of the "geometric" naming of the objects see Section 6 of [54]). The formalism can be put in the matrix model form, in which case MHSYM action (7) becomes…”

“…We present here four types of matter that can be coupled in a way that respects MHS symmetry. The first three, described in Sections 2.3.1-2.3.3, were already studied in [54].…”

“…It is possible to couple an ordinary relativistic spacetime field ψ r (x) to the MHS vielbein (the subscript r denotes both Lorentz and internal indices). This approach was originally introduced in [46,47] for the case of the Klein-Gordon field and in [51,53] for the case of the Dirac field, and generalized to all spins in [54]. The procedure goes as follows.…”

“…This approach, although leading to an effective action with an infinite number of vertices whose explicit form can hardly be determined beyond the cases with lowest number of interacting fields, has the crucial advantages of assuming a very simple gauge symmetry in a form analogous to the one of Yang-Mills (YM) gauge models and inducing a natural Moyal-product structure in terms of which the HS couplings can be compactly written in terms of master fields in a non-commutative space including the ordinary spacetime and an auxiliary space of the same dimension. The simplest model in the class of theories possessing this symmetry is the Moyal-higher-spin YM (MHSYM) one studied in [52][53][54], which naturally allows for different backgrounds, and is an ideal setup for the study of classical solutions and the description of a HS geometry. Furthermore MHSYM can be considered as the most natural starting point to study the quantization of this class of theories, which is in principle made possible by the exact knowledge of the gauge symmetry.…”