Lie and Leibniz Algebras of Lower-Degree Conservation Laws

TL;DR

This paper explores the algebraic structure of conservation laws in gauge theories, revealing that they form Leibniz algebras rather than Lie algebras, with implications demonstrated through Yang-Mills and Einstein gravity examples.

Contribution

It establishes that the algebra of asymptotic charges in gauge theories is fundamentally Leibniz, not Lie, and provides a general construction with illustrative examples.

Findings

Conservation laws form Leibniz algebras in gauge theories.

Under certain conditions, Leibniz algebras reduce to Lie algebras.

The construction is exemplified in Yang-Mills and Einstein gravity.

Abstract

A relationship between the asymptotic and lower-degree conservation laws in (non-)linear gauge theories is considered. We show that the true algebraic structure underlying asymptotic charges is that of Leibniz rather than Lie. The Leibniz product is defined through the derived bracket construction for the natural Poisson brackets and the BRST differential. Only in particular, though not rare, cases that the Poisson brackets of lower-degree conservation laws vanish modulo central charges, the corresponding Leibniz algebra degenerates into a Lie one. The general construction is illustrated by two standard examples: Yang-Mills theory and Einstein's gravity.