Lorentz Covariant Lagrangians of Self-dual Gauge Fields

TL;DR

This paper develops a systematic method to covariantize non-covariant Lagrangians of self-dual gauge fields using the PST formulation, revealing the structure of extra local symmetries and establishing a no-go theorem for certain decompositions.

Contribution

It extends the PST method to derive covariant Lagrangians for self-dual gauge fields in various spacetime decompositions, including a proof of limitations for the 2+4 case.

Findings

Covariant Lagrangians are derived for 6=1+5 and 6=3+3 decompositions.

The extra local symmetry can be expressed linearly in the field strength for some decompositions.

A no-go theorem shows the symmetry cannot be simplified similarly in the 6=2+4 decomposition.

Abstract

We extend the method of PST formulation to find a systematic way to covariantize several non-covariant Lagrangians of self-dual gauge fields. We derive in detail the necessary basic formulas which are used to prove the existence of extra local symmetry that allows us to gauge fix the auxiliary fields therein and non-covariant formulations are restored. We see that, the extra local symmetry in the PST and PSST formulations, which describe the covariant Lagrangians in the 6D decomposition of $6=1+5$ and $6=3+3$ respectively, can be expressed as a simple linear form in the field strength. However, although in this paper we have found the covariant Lagrangians in the other decomposition of $6=2+4$, the extra local symmetry of the gauge field cannot be expressed as a simple linear form in the field strength. We present a no-go theorem to prove this specific property. We also find other covariant Lagrangians with more complex decomposition of spacetime.