[Paper Review] The division map of principal bundles with groupoid structure and generalized gauge transformations

Carlo A. Rossi|ArXiv.org|Jan 15, 2004

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper generalizes the concept of generalized gauge transformations to principal bundles with structure groupoid $Γ$, extending the role of the division map $φ_P$—previously used in ordinary principal $G$-bundles—to the framework of Lie groupoids. It introduces the groupoid of Hilsum–Skandalis generalized gauge transformations, showing that these transformations form a well-defined groupoid under composition, with invertible morphisms and a group structure on isotropy groups, thereby establishing a categorical and geometric foundation for gauge theory in non-abelian and non-locally trivial settings.

ABSTRACT

Motivated by the computations done in \cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category $\mathsf{Bun}_{G,M}$ of principal $G$-bundles over a given manifold $M$, I develop in this paper the same ideas for the more general case of {\em principal $\calG$-bundles or principal bundles with structure groupoid $\calG$}, where now $\calG$ is a Lie groupoid in the sense of \cite{Moer2}. Most of the concepts introduced in \cite{C1} can be translated almost verbatim in the framework of principal bundles with structure groupoid $\calG$; in particular, the key r�le for the construction of generalized gauge transformations is again played by (the equivalent in the framework of principal bundles with groupoid structure of) the division map $f_P$. Of great importance are also the generalized conjugation in a groupoid and the concept of (twisted) equivariant maps between groupoid-spaces.

Motivation & Objective

• To extend the theory of generalized gauge transformations from principal $G$-bundles to principal bundles with structure groupoid $\mathcal{G}$, where $\mathcal{G}$ is a Lie groupoid.
• To generalize the role of the division map $\phi_P$ in characterizing free and transitive groupoid actions on fibers of principal $\mathcal{G}$-bundles.
• To define and study the groupoid of Hilsum–Skandalis generalized gauge transformations, establishing its structure and invariance under bundle isomorphisms.
• To show that the space of generalized gauge transformations between Hilsum–Skandalis morphisms inherits a groupoid structure via a well-defined product $\star$, compatible with $\mathcal{G}$-invariance.
• To prove that the isotropy groups of this generalized gauge groupoid are isomorphic to the $\mathcal{H}$-invariant smooth maps from the bundle to $\mathcal{H}$, thus recovering the gauge group as a special case.

Proposed method

• The paper defines principal bundles with structure groupoid $\mathcal{G}$ as smooth surjective submersions $\pi: P \to M$ equipped with a free and transitive $\mathcal{G}$-action on each fiber, generalizing the notion of principal $G$-bundles.
• It introduces the division map $\phi_P: P \times_M P \to \mathcal{G}$, which encodes the groupoid action and generalizes the classical division map for $G$-bundles.
• The paper constructs generalized gauge transformations as $\mathcal{G}$-invariant, $\mathcal{H}$-equivariant maps from the fiber product $P_1 \odot P_2$ to the groupoid $\mathcal{H}$, using the product $\star$ defined by $ (K_{23} \star K_{12})(p_1,p_3) = K_{23}(p_2,p_3)K_{12}(p_1,p_2) $.
• It proves that the product $\star$ is associative and independent of the choice of intermediate point $p_2$, ensuring well-defined composition in the groupoid of generalized gauge transformations.
• It establishes that the inverse of any generalized gauge transformation is given by the inverse in $\mathcal{H}$ of the division map, ensuring invertibility and groupoid structure.
• It defines the groupoid $C^{\infty,\mathcal{H}^2}_{\mathcal{G}}$ whose objects are Hilsum–Skandalis morphisms and whose morphisms are $\mathcal{G}$-invariant $\mathcal{H}$-valued maps on fiber products, showing it is isomorphic to the category $\mathsf{HS}_{\mathcal{G},\mathcal{H}}$.

Experimental results

Research questions

• RQ1 How can the concept of generalized gauge transformations be extended from principal $G$-bundles to principal bundles with structure groupoid $\mathcal{G}$, where $\mathcal{G}$ is a Lie groupoid?
• RQ2 What is the role of the division map $\phi_P$ in characterizing the structure of principal $\mathcal{G}$-bundles and enabling the construction of generalized gauge transformations?
• RQ3 Can the space of generalized gauge transformations between Hilsum–Skandalis morphisms be endowed with a well-defined groupoid structure under composition?
• RQ4 How does the $\mathcal{G}$-invariance of the transformation product $\star$ ensure consistency across different choices of intermediate points in the fiber product?
• RQ5 What is the relationship between the isotropy group of the generalized gauge groupoid and the space of $\mathcal{H}$-invariant smooth maps from a bundle to $\mathcal{H}$?

Key findings

• The groupoid of Hilsum–Skandalis generalized gauge transformations is well-defined, with morphisms closed under the product $\star$, which is associative and independent of intermediate point choices.
• The product $\star$ of two generalized gauge transformations is $\mathcal{G}$-invariant, ensuring that the composition descends to a morphism in the category of Hilsum–Skandalis morphisms.
• Every generalized gauge transformation is invertible, with the inverse given by the inverse in $\mathcal{H}$ of the division map, ensuring the groupoid structure.
• The isotropy group at any object $P$ in the generalized gauge groupoid is isomorphic to the space $C^{\infty}_{\mathcal{G}}(P,\mathcal{H})^{\mathcal{H}}$, which is the Hilsum–Skandalis gauge group of $P$.
• The groupoid $C^{\infty,\mathcal{H}^2}_{\mathcal{G}}$ is isomorphic to the category $\mathsf{HS}_{\mathcal{G},\mathcal{H}}$, confirming that all morphisms are invertible and the structure is a genuine groupoid.
• The construction generalizes the classical gauge group and transformation theory to the non-abelian, non-locally trivial setting of Lie groupoid actions, providing a categorical framework for generalized gauge theory.