# Parallel transport in principal 2-bundles

@article{Waldorf2017ParallelTI, title={Parallel transport in principal 2-bundles}, author={Konrad Waldorf}, journal={arXiv: Differential Geometry}, year={2017} }

A nice differential-geometric framework for (non-abelian) higher gauge theory is provided by principal 2-bundles, i.e. categorified principal bundles. Their total spaces are Lie groupoids, local trivializations are kinds of Morita equivalences, and connections are Lie-2-algebra-valued 1-forms. In this article, we construct explicitly the parallel transport of a connection on a principal 2-bundle. Parallel transport along a path is a Morita equivalence between the fibres over the end points, and…

## 10 Citations

### A global perspective to connections on principal 2-bundles

- MathematicsForum Mathematicum
- 2018

For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action…

### A Global Geometric Approach to Parallel Transport of Strings in Gauge Theory

- Mathematics
- 2019

This paper is motivated by recent development of higher gauge theory. Different from its style of using higher category theory, we try to describe the concept of higher parallel transport within…

### Gerbes in Geometry, Field Theory, and Quantisation

- MathematicsComplex Manifolds
- 2021

Abstract This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes…

### Parallel 2-transport and 2-group torsors

- Mathematics
- 2018

We provide a new perspective on parallel 2-transport and principal 2-group bundles with 2-connection. We define parallel 2-transport as a 2-functor from the thin fundamental 2-groupoid to the…

### Operational total space theory of principal 2-bundles I: Operational geometric framework

- Mathematics
- 2019

### Operational total space theory of principal 2-bundles II: 2-connections and 1- and 2-gauge transformations

- MathematicsJournal of Geometry and Physics
- 2020

### Two-dimensional algebra in lattice gauge theory

- MathematicsJournal of Mathematical Physics
- 2019

We provide a visual and intuitive introduction to effectively calculating in 2-groups along with explicit examples coming from non-abelian 1- and 2-form gauge theory. In particular, we utilize string…

### The higher Riemann-Hilbert correspondence and principal 2-bundles

- Mathematics
- 2020

In this note we show that the holonomies for higher local systems provided by the higher Riemann-Hilbert correspondence coincide with those associated to principal $2$-bundles where both formalisms…

### The First Pontryagin Class of a Quadratic Lie 2-Algebroid

- MathematicsCommunications in Mathematical Physics
- 2018

In this paper, we first give a detailed study on the structure of a transitive Lie 2-algebroid and describe a transitive Lie 2-algebroid using a morphism from the tangent Lie algebroid T M to a…

### Pushforwards and Gauge Transformations for Categorical Connections.

- Mathematics
- 2020

We construct and study pushforwards of categorical connections on categorical principal bundles. Applying this construction to the case of decorated path spaces in principal bundles, we obtain a…

## References

SHOWING 1-10 OF 14 REFERENCES

### A global perspective to connections on principal 2-bundles

- MathematicsForum Mathematicum
- 2018

For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action…

### Higher gauge theory

- Mathematics
- 2005

Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings)…

### Principal 2-bundles and their gauge 2-groups

- Mathematics
- 2008

Abstract In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Čech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued…

### Nonabelian Bundle Gerbes, Their Differential Geometry and Gauge Theory

- Physics
- 2005

Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied…

### Local theory for 2-functors on path 2-groupoids

- Mathematics
- 2017

This article is concerned with 2-functors defined on the path 2-groupoid of a smooth manifold. We set up a procedure to extract local data of such 2-functors, similar to the extraction of transition…

### Connections on non-abelian Gerbes and their Holonomy

- Mathematics
- 2008

We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing…

### Bundle gerbes

- Mathematics
- 1994

. Just as C × principal bundles provide a geometric realisation of two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a geometric realisation of three dimensional integral…

### Smooth functors vs. differential forms

- Mathematics
- 2008

We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a…