Higher dualisations of linearised gravity and the $$ {A}_1^{+++} $$ algebra

Abstract

The non-linear realisation based on $$ {A}_1^{+++} $$ A 1 + + + is known to describe gravity in terms of both the graviton and the dual graviton. We extend this analysis at the linearised level to find the equations of motion for the first higher dual description of gravity that it contains. We also give a systematic method for finding the additional fields beyond those in the non-linear realisation that are required to construct actions for all of the possible dual descriptions of gravity in the non-linear realisation. We show that these additional fields are closely correlated with the second fundamental representation of $$ {A}_1^{+++} $$ A 1 + + + .

Dualisation of free fields

“…it contains all the lower order consequences of (4). Besides the Noether identities (8), the involutive closure (10) by construction enjoys the extra gauge identities…”

“…As a first step to constructing dual formulation for the original free field theory, let us assume that the set of consequences (9), being considered by itself, irrespectively to entire system (10), admit a more wide gauge symmetry than the original system:…”

“…, n, are assumed to be arbitrary functions of x. Once the fields are tensors, and the involutive closure of EoMs (10) is a covariant system, the gauge parameters have to be tensors.…”

“…Let us further assume that consequences τ a (9), being considered irrespectively to the complete system (10), form a topological subsystem of EoMs in the sense that it does not have any local DoF. Given the complete sequences of gauge symmetries and identities of the involutive system, one can immediately count the DoF [12], so this assumption can be always verified.…”

“…Various cases are long known of the duality of this type, see, e.g. [1]- [10] and references therein. Another option is that dual fields can be connected by differential relations like strength tensors and their potentials, and these relations are not invertible at the local level even upon gauge fixing.…”