Conjugacy classes of gauge groups

R. Piccinini

Abstract

Let ξ be a smooth, principal G-bundle over a differentiable manifold B representing a physical theory; an equivariant, fibre preserving bundle automorphism of ξ gives rise to a new bundle with identical physical content and thus, such an automorphism is a symmetry of the physical theory. The set G(ξ) of all such automorphisms (called gauge transformations) has the structure of a topological group, called gauge group of ξ. A direct analysis and classification of gauge groups as algebraic objects is rather complicated; a simpler way seems to be the study of the conjugacy classes of the gauge groups of different bundles within a larger group of symmetries (depending on B). A first interesting result is the following: let ZG be the centre of the group G; then, the set E_{ZG}(B) of all equivalence classes of principal ZG-bundles over B is an abelian group acting on the set E_{G}(B) of all equivalence classes of principal G-bundles over B:

⊗ : E_{ZG}(B) × E_{G}(B) → E_{G}(B)

Then, G(ξ′) and G(ξ) are conjugate if, and only if, there exists λ ∈ E_{ZG}(B) such that ξ′ ≅ λ ⊗ ξ.

The group I(ξ) of all λ ∈ E_{ZG}(B) such that λ⊗ξ ≅ ξ (the isotropy group of ξ) gives good informations about the conjugacy class of G(ξ); the isotropy group I(ξ) is known, for example, for n-bundles over real projective n-spaces and plane bundles over compact, connected surfaces.

I plan to give a general overview of these concepts and some of their connections with homotopy theory, without resorting to highly technical details.