Commutative diagrams of Sp(2) Symmetry

TL;DR

This paper explores the geometric and topological structures of Sp(2) symmetry in gauge theories, emphasizing the construction of commutative diagrams to understand cocycle conditions within the covariant quantization framework.

Contribution

It introduces a topological approach to gauge theories with Sp(2) symmetry, focusing on commutative diagrams and cocycle conditions in the covariant quantization scheme.

Findings

Topological structures for gauge theories with Sp(2) symmetry are developed.

Commutative diagrams for cocycle conditions are constructed.

Geometrical interpretation of gauge theories via fibre bundles is clarified.

Abstract

It is well known that BRST symmetry plays a fundamental role in constructing quantum gauge theories. Yet, at the classical level, it constitutes the modern language to study constrained systems. First, this letter reviews the Sp(2) covariant quantisation of gauge theories, so that the geometrical interpretation of gauge theories in terms of quasi-principal fibre bundles Q(Ms, Gs) is the main scenario. It is then described the Sp(2) algebra for ordinary Yang-Mills theory. Some basic set theory and topology terms are reviewed to proceed then to use theorems on left inverse and right inverse of functions together with groups (Ms, Gs) to have triangular compositions of functions on manifolds. Henceforth, it is so the purpose of this letter to present topological structures to gauge theories, in particular to construct commutative diagrams for cocycle conditions of the Sp(2) symmetry.