A connection between linearized Gauss-Bonnet gravity and classical electrodynamics II: Complete dual formulation

Abstract

In a recent publication a procedure was developed which can be used to derive completely gauge invariant models from general Lagrangian densities with $N$ order of derivatives and $M$ rank of tensor potential. This procedure was then used to show that unique models follow for each order, namely classical electrodynamics for $N = M = 1$ and linearized Gauss-Bonnet gravity for $N = M = 2$. In this article, the nature of the connection between these two well explored physical models is further investigated by means of an additional common property; a complete dual formulation. First we give a review of Gauss-Bonnet gravity and the dual formulation of classical electrodynamics. The dual formulation of linearized Gauss-Bonnet gravity is then developed. It is shown that the dual formulation of linearized Gauss-Bonnet gravity is analogous to the homogenous half of Maxwell's theory; both have equations of motion corresponding to the (second) Bianchi identity, built from the dual form of their respective field strength tensors. In order to have a dually symmetric counterpart analogous to the non-homogenous half of Maxwell's theory, the first invariant derived from the procedure in $N = M = 2$ can be introduced. The complete gauge invariance of a model with respect to Noether's first theorem, and not just the equation of motion, is a necessary condition for this dual formulation. We show that this result can be generalized to the higher spin gauge theories, where the spin-$n$ curvature tensors for all $N = M = n$ are the field strength tensors for each $n$. These completely gauge invariant models correspond to the Maxwell-like higher spin gauge theories whose equations of motion have been well explored in the literature.