Parallel transport and holonomy of gerbes

Parallel transport and holonomy are well-known as notions in the tangent bundle of a Riemannian manifold. More abstractly, parallel transport and holonomy can be studied in vector bundles or principal bundles with connections. In the context of higher-categorical geometry I study analogous notions for gerbes and so-called 2-vector bundles. Here, parallel transport and holonomy are not evaluated along path but along surfaces. Completely new aspects arise, such as the fact that while every line is orientable, there exist surfaces that are not orientable, for instance the Klein Bottle. My work about higher-categorical parallel transport and holonomy concerns foundational aspects, e.g. the precise formulation of what parallel transport along a surface actually is, and the comparison of the various different versions.

Below are some articles and manuscripts of talks about this topic.

Articles associated with this topic

  • Smooth Functorial Field Theories from B-Fields and D-Branes
    together with Severin Bunk
    J. Homotopy Relat. Struct. 16.1 (2021): 75-153
  • Parallel transport in principal 2-bundles
    Higher Structures 2(1):57-115, 2018
  • A global perspective to connections on principal 2-bundles
    Forum Math. 30 (2017), no. 4, 809-843
  • Local Theory for 2-Functors on Path 2-Groupoids
    together with Urs Schreiber
    J. Homotopy Relat. Struct. (2016) 1-42
  • Connections on non-abelian Gerbes and their Holonomy
    together with Urs Schreiber
    Theory Appl. Categ., Vol. 28, 2013, No. 17, pp 476-540
  • Smooth Functors vs. Differential Forms
    together with Urs Schreiber
    Homology, Homotopy Appl., 13(1), 143-203 (2011)
  • Parallel Transport and Functors
    together with Urs Schreiber
    J. Homotopy Relat. Struct. 4, 187-244 (2009)

Talks associated with this topic

  • Parallel Transport Functors of Principal Bundles and (non-abelian) Bundle Gerbes
    Conference "Principal Bundles, Gerbes and Stacks", Physikzentrum Bad Honnef, June 2007
  • Parallel Transport and Functors
    Conference "Categories in Geometry and mathematical Physics", Mediterranean Institute For Life Sciences, September 2007
  • Transport Functors and Connections on Gerbes
    Seminar "Topology", University of California at Berkeley, August 2008
    Notes Part 1   Notes Part 2  
  • Smooth Functors for higher-dimensional Parallel Transport
    Workshop "Smooth Structures in Logic, Category Theory and Physics", Ottawa University, May 2009
  • An introduction to higher parallel transport
    Seminar "Algebraic and combinatorial perspectives in the mathematical sciences", online, September 2020
    Presentation   Video