Transgression to Loop Spaces

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

  • The stringor bundle
  • Smooth Functorial Field Theories from B-Fields and D-Branes
    together with Severin Bunk
    J. Homotopy Relat. Struct. 16.1 (2021): 75-153
  • Transgression of D-branes
    together with Severin Bunk
    Adv. Theor. Math. Phys., Vol. 25, No. 5 (2021), pp. 1095-1198.
  • Transgressive loop group extensions
    Math. Z. 286(1) 325-360, 2017
  • 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
  • Lifting Problems and Transgression for Non-Abelian Gerbes
    together with Thomas Nikolaus
    Adv. Math. 242 (2013) 50-79
  • Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles
    Adv. Math. 231 (2012), 3445-3472
  • A Loop Space Formulation for Geometric Lifting Problems
    J. Aust. Math. Soc. 90, 129-144 (2011)
  • 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
  • Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps
    Cah. Topol. Géom. Différ. Catég., 2012, Vol. LIII, 162-210
  • Multiplicative Bundle Gerbes with Connection
    Differential Geom. Appl. 28(3), 313-340 (2010)

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
  • Abelian gauge theories on loop spaces and their regression
    Workshop "Higher Gauge Theory, TQFT and Quantum Gravity", Instituto Superior Técnico Lisboa, February 2011
  • A loop space formulation for the geometry of abelian gerbes
    Conference "Analysis, Geometry, and Quantum Field Theory", Universität Potsdam, October 2011
  • 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
  • Transgressive central extensions of loop groups
    Conference "Colloquium on Algebras and Representations - Quantum 2016", Universidad Nacional de Córdoba, March 2016
  • String geometry and spin geometry on loop spaces
    Parallel session "Mathematical aspects of string theory and string geometry", Friedrich-Schiller-Universität Jena, July 2016
  • String connections and loop spaces
    Workshop "Loop spaces, supersymmetry and index theory", Nankai University at Tianjin, July 2017
  • Fusion in loop spaces
    Workshop "Geometric Quantization", Banff International Research Station for Mathematical Innovation and Discovery, April 2018
  • Transgression of higher structures to loop spaces
    Workshop "Loop Space and Higher Category", online, December 2022
  • The stringor bundle
    Conference "Geometries from Strings and Fields", Galileo Galilei Institute, July 2023