Higher spins from exotic dualisations

Nicolas Boulanger,Victor Lekeu

doi:10.1007/jhep03(2021)171

Nicolas Boulanger, Victor Lekeu

Open Access

https://doi.org/10.1007/jhep03(2021)171

Copy DOI

Abstract

At the free level, a given massless field can be described by an infinite number of different potentials related to each other by dualities. In terms of Young tableaux, dualities replace any number of columns of height hi by columns of height D − 2 − hi, where D is the spacetime dimension: in particular, applying this operation to empty columns gives rise to potentials containing an arbitrary number of groups of D − 2 extra antisymmetric indices. Using the method of parent actions, action principles including these potentials, but also extra fields, can be derived from the usual ones. In this paper, we revisit this off-shell duality and clarify the counting of degrees of freedom and the role of the extra fields. Among others, we consider the examples of the double dual graviton in D = 5 and two cases, one topological and one dynamical, of exotic dualities leading to spin three fields in D = 3.

Highlights

Consider mixed-symmetry gauge fields of arbitrary symmetry type. (This includes in particular the case of higher-spin fields, for which each column has one box)
One could dualise on any number of columns of height hi, replacing them with columns of height D − 2 − hi [3, 10,11,12], or dualise on empty columns to add an arbitrary number of columns of height D − 2 to the left of the Young diagram [3, 9, 13, 14]
As a consequence, when computing the Euler-Lagrange equations of motion we find that the method of parent actions gives a version of the on-shell duality relations that is invariant under these entangled gauge symmetries

Summary

The two column case

We first consider the simplest case of a massless scalar field φ in three dimensions. Given the result of the previous examples, one can guess the structure of an action giving these equations: the difference of the conventional Lagrangian for the C-field and the Fierz-Pauli action, accompanied by a gauge-invariant cross-term, S[Cab|cd, hab] = d5x α L[2,2] − LFP + βLcross (2.57). We show that it is this structure that comes out of the off-shell d√uality procedure of [13] for the double-dual graviton, with the result α = −1 and β = − 3/2 It provides a dual action in terms of a gauge potential Dabc|de which is antisymmetric in both groups of indices, but satisfying no other condition.

Topological case: spin 2 – spin 3

Dynamical case: spin 1 – spin 3

Conclusions