The Action for Twisted Self-Duality

TL;DR

This paper systematically derives an action principle for twisted self-duality in gauge theories, extending to various fields and couplings, ensuring Lorentz invariance and local gravity coupling, with a focus on Hamiltonian formalism.

Contribution

It provides a Hamiltonian derivation of the twisted self-duality action, extending previous ansatzes and analyzing Lorentz invariance and couplings in a unified framework.

Findings

Recovered known actions for pure Maxwell case

Extended the action to include Chern-Simons couplings with new results

Provided a Hamiltonian proof of Lorentz invariance and local coupling to gravity

Abstract

One may write the Maxwell equations in terms of two gauge potentials, one electric and one magnetic, by demanding that their field strengths should be dual to each other. This requirement is the condition of twisted self-duality. It can be extended to p-forms in spacetime of D dimensions, and it survives the introduction of a variety of couplings among forms of different rank, and also to spinor and scalar fields, which emerge naturally from supergravity. In this paper we provide a systematic derivation of the action principle, whose equations of motion are the condition of twisted self-duality. The derivation starts from the standard Maxwell action, extended to include the aforementioned couplings, and proceeds via the Hamiltonian formalism through the resolution of Gauss' law. In the pure Maxwell case we recover in this way an action that had been postulated by other authors, through an ansatz based on an action given earlier by us for untwisted self-duality. Those authors also extended their ansatz to include Chern-Simons couplings. In that case, we find a different result. The derivation from the standard extended Maxwell action implies of course that the theory is Lorentz-invariant and can be locally coupled to gravity. Nevertherless we include a direct compact Hamiltonian proof of these properties, which is based on the surface-deformation algebra. The symmetry in the dependence of the action on the electric and magnetic variables is manifest, since they appear as canonical conjugates. Spacetime covariance, although present, is not manifest.