In the simplest case, parallel transport and holonomy are known as concepts in the tangent bundle of a Riemannian manifold. More abstractly, one can study them on vector bundles or principal fiber bundles with connection. Within the framework of higher-categorical geometry, I am investigating analogous concepts for gerbes and 2-vector bundles. Here, parallel transport and holonomy are evaluated not only along curves but also along pieces of surfaces. The study of these generalizations is motivated significantly by gauge fields in string theory, described through connections on sheaves, and coupled to charged strings through their parallel transport.
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
arxiv:1911.09990Parallel transport in principal 2-bundles
Higher Structures 2(1):57-115, 2018
arxiv:1704.08542A global perspective to connections on principal 2-bundles
Forum Math. 30 (2017), no. 4, 809-843
arxiv:1608.00401Local Theory for 2-Functors on Path 2-Groupoids
together with Urs Schreiber
J. Homotopy Relat. Struct. (2016) 1-42
arxiv:1303.4663Connections on non-abelian Gerbes and their Holonomy
together with Urs Schreiber
Theory Appl. Categ., Vol. 28, 2013, No. 17, pp 476-540
arxiv:0808.1923Smooth Functors vs. Differential Forms
together with Urs Schreiber
Homology, Homotopy Appl., 13(1), 143-203 (2011)
arxiv:0802.0663Parallel Transport and Functors
together with Urs Schreiber
J. Homotopy Relat. Struct. 4, 187-244 (2009)
arxiv:0705.0452
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
NotesParallel Transport and Functors
Conference "Categories in Geometry and mathematical Physics", Mediterranean Institute For Life Sciences, September 2007
NotesTransport Functors and Connections on Gerbes
Seminar "Topology", University of California at Berkeley, August 2008
Notes Part 1 Notes Part 2Smooth Functors for higher-dimensional Parallel Transport
Workshop "Smooth Structures in Logic, Category Theory and Physics", Ottawa University, May 2009
NotesAn introduction to higher parallel transport
Seminar "Algebraic and combinatorial perspectives in the mathematical sciences", online, September 2020
Presentation Video