Transgression transforms a gerbe with connection over a manifold M into a line bundle with connection over the free loop space LM, and so establishes a functorial relation between higher-categorical geometry on M and ordinary geometry on LM. In 2-dimensional field theories, for which connections on gerbes represent the gauge fields, the corresponding line bundles play the role of prequantum line bundles, and let us look at the loop space as a kind of symplectic manifold.
For my research the most interesting aspect of transgression is that all line bundles in the image of transgression carry interesting additional structure: so-called fusion products, and equivariance under thin homotopies between loops. These additional structures remember information of the given gerbe that would be lost upon looking at the line bundle alone. Among other things, they admit to invert transgression, and so to go back from infinite-dimensional geometry of LM to higher-categorical geometry over M.
Below are some articles and manuscripts of talks about this topic.
Articles associated with this topic
- A representation of the string 2-group
arxiv:2206.09797 - Smooth Functorial Field Theories from B-Fields and D-Branes
together with Severin Bunk
J. Homotopy Relat. Struct. 16.1 (2021): 75-153
arxiv:1911.09990 - Transgression of D-branes
together with Severin Bunk
Adv. Theor. Math. Phys., Vol. 25, No. 5 (2021), pp. 1095-1198.
arxiv:1808.04894 - Transgressive loop group extensions
Math. Z. 286(1) 325-360, 2017
arxiv:1502.05089 - A Construction of String 2-Group Models using a Transgression-Regression Technique
Analysis, Geometry and Quantum Field Theory, edited by C. L. Aldana, M. Braverman, B. Iochum, and C. Neira-Jiménez, volume 584 of Contemp. Math., pages 99-115, AMS, 2012
arxiv:1201.5052 - Lifting Problems and Transgression for Non-Abelian Gerbes
together with Thomas Nikolaus
Adv. Math. 242 (2013) 50-79
arxiv:1112.4702 - Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles
Adv. Math. 231 (2012), 3445-3472
arxiv:1109.0480 - A Loop Space Formulation for Geometric Lifting Problems
J. Aust. Math. Soc. 90, 129-144 (2011)
arxiv:1007.5373 - Transgression to Loop Spaces and its Inverse, II: Gerbes and Fusion Bundles with Connection
Asian J. Math., Vol. 20, No. 1 (2016), pp. 59-116
arxiv:1004.0031 - Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps
Cah. Topol. Géom. Différ. Catég., 2012, Vol. LIII, 162-210
arxiv:0911.3212 - Multiplicative Bundle Gerbes with Connection
Differential Geom. Appl. 28(3), 313-340 (2010)
arxiv:0804.4835
Talks associated with this topic
- Transgression of Gerbes to Loop Spaces
Workshop "Higher Structures in Topology and Geometry IV", Georg-August-Universität Göttingen, June 2010
Notes - Abelian gauge theories on loop spaces and their regression
Workshop "Higher Gauge Theory, TQFT and Quantum Gravity", Instituto Superior Técnico Lisboa, February 2011
Notes - A loop space formulation for the geometry of abelian gerbes
Conference "Analysis, Geometry, and Quantum Field Theory", Universität Potsdam, October 2011
Notes - Differential string classes and loop spaces
Workshop "Differential Cohomologies", The Graduate Center, CUNY, August 2014
Video Notes - String structures and supersymmetric sigma models
Program "Higher structures in string theory and quantum field theory", Erwin-Schrödinger-Institut für Mathematische Physik, December 2015
Notes - Transgressive central extensions of loop groups
Conference "Colloquium on Algebras and Representations - Quantum 2016", Universidad Nacional de Córdoba, March 2016
Notes - String geometry and spin geometry on loop spaces
Parallel session "Mathematical aspects of string theory and string geometry", Friedrich-Schiller-Universität Jena, July 2016
Notes - String connections and loop spaces
Workshop "Loop spaces, supersymmetry and index theory", Nankai University at Tianjin, July 2017
Presentation - Fusion in loop spaces
Workshop "Geometric Quantization", Banff International Research Station for Mathematical Innovation and Discovery, April 2018
Video - Transgression of higher structures to loop spaces
Workshop "Loop Space and Higher Category", online, December 2022
Presentation