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On gauge theory and parallel transport in principal 2-bundles over Lie groupoids

Adittya Chaudhuri

Abstract

We investigate an interplay between some ideas in traditional gauge theory and certain concepts in fibered categories. We accomplish this by introducing a notion of a principal Lie 2-group bundle over a Lie groupoid and studying its connection structures, gauge transformations, and parallel transport. We obtain a Lie 2-group torsor version of the one-one correspondence between fibered categories and pseudofunctors. This results in a classification of our principal 2-bundles based on their underlying fibration structures. This allows us to extend a class of our principal 2-bundles to be defined over differentiable stacks presented by the base Lie groupoids. We construct a short exact sequence of VB-groupoids, namely, the 'Atiyah sequence' associated to our principal 2-bundles. Splitting and splitting up to a natural isomorphism of our Atiyah sequence, respectively, gives us notions of 'strict connections' and 'semi-strict connections' on our principal 2-bundles. We describe such connections in terms of Lie 2-algebra valued 1-forms on the total Lie groupoids. The underlying fibration structure of our 2-bundle provides an existence criterion for strict and semi-strict connections. We study the action of the 2-group of gauge transformations on the groupoid of strict and semi-strict connections, and interestingly, we observe an extended symmetry of semi-strict connections. We demonstrate an interrelationship between `differential geometric connection-induced horizontal path lifting property in traditional principal bundles' and the `category theoretic cartesian lifting of morphisms in fibered categories' by developing a theory of connection-induced parallel transport along a particular class of Haefliger paths in the base Lie groupoid of our principle 2-bundles. Finally, we employ our results to introduce a notion of parallel transport along Haefliger paths in the setup of VB-groupoids.

On gauge theory and parallel transport in principal 2-bundles over Lie groupoids

Abstract

We investigate an interplay between some ideas in traditional gauge theory and certain concepts in fibered categories. We accomplish this by introducing a notion of a principal Lie 2-group bundle over a Lie groupoid and studying its connection structures, gauge transformations, and parallel transport. We obtain a Lie 2-group torsor version of the one-one correspondence between fibered categories and pseudofunctors. This results in a classification of our principal 2-bundles based on their underlying fibration structures. This allows us to extend a class of our principal 2-bundles to be defined over differentiable stacks presented by the base Lie groupoids. We construct a short exact sequence of VB-groupoids, namely, the 'Atiyah sequence' associated to our principal 2-bundles. Splitting and splitting up to a natural isomorphism of our Atiyah sequence, respectively, gives us notions of 'strict connections' and 'semi-strict connections' on our principal 2-bundles. We describe such connections in terms of Lie 2-algebra valued 1-forms on the total Lie groupoids. The underlying fibration structure of our 2-bundle provides an existence criterion for strict and semi-strict connections. We study the action of the 2-group of gauge transformations on the groupoid of strict and semi-strict connections, and interestingly, we observe an extended symmetry of semi-strict connections. We demonstrate an interrelationship between `differential geometric connection-induced horizontal path lifting property in traditional principal bundles' and the `category theoretic cartesian lifting of morphisms in fibered categories' by developing a theory of connection-induced parallel transport along a particular class of Haefliger paths in the base Lie groupoid of our principle 2-bundles. Finally, we employ our results to introduce a notion of parallel transport along Haefliger paths in the setup of VB-groupoids.

Paper Structure

This paper contains 92 sections, 99 theorems, 256 equations, 1 figure.

Table of Contents

  1. Introduction and overview
  2. Organization of the thesis
  3. Classical Set-up
  4. Preliminaries
  5. Principal 2-bundles over Lie groupoids and their characterizations
  6. Connection structures and gauge transformations on a principal 2-bundle over a Lie groupoid
  7. Parallel transport on quasi-principal 2-bundles and associated VB-groupoids
  8. Future directions of research
  9. Notations and conventions
  10. Classical Set-up
  11. A principal bundle, its Atiyah sequence and its gauge group
  12. Fibre bundles
  13. Atiyah sequence associated to a principal bundle
  14. Gauge group of a principal bundle
  15. Connection structures on a principal bundle
  16. ...and 77 more sections

Key Result

theorem 1

For a Lie crossed module $(G,H,\tau, \alpha)$ and a Lie groupoid $\mathbb{X}$, the groupoid $\rm{Bun}_{\rm{quasi}}(\mathbb{X}$, $[H \rtimes_{\alpha}G \rightrightarrows G])$ is equivalent to the groupoid ${\rm{Pseudo}} (\mathbb{X}, (G,H,\tau, \alpha) )$, where $\rm{Bun}_{\rm{quasi}}(\mathbb{X}$, $[H

Theorems & Definitions (411)

  • theorem 1: \ref{['Main Theorem 1']}
  • Proposition 2: \ref{['Prop:AtiyahLie2gpd']}
  • theorem 3: \ref{['strict connection=strict forms']}
  • theorem 4: \ref{['Theorem: Parallel transport on 2-bundles']}
  • Definition 5
  • Definition 6
  • Definition 7
  • Remark 8
  • Definition 9
  • Definition 10
  • ...and 401 more