*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

*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