String geometry is a relatively new research area on the intersection between Algebraic Topology, Differential Geometry, and Homotopy Theory. It provides a mathematical basis for the description of supersymmetry in two-dimensional quantum field theories; from this point of view string geometry is for string theory as spin geometry is for quantum mechanics.

There are essentially two approaches to string geometry: infinite-dimensional analysis on the configuration space of the strings, or higher-categorical analysis on the target space of the strings. The configuration space is the loop space of the target space, and both approaches should be related by a transgression process.

Infinite-dimensional analysis on the loop space leads to long open questions such that how to define a Dirac operator on the loop space, and on which kind of representation this operator could act on. In my work I try to understand these questions via higher-categorical geometry on the target space under transgression.

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

## Articles associated with this topic

*Connes fusion of spinors on loop space*

arxiv:2012.08142*Smooth Fock bundles, and spinor bundles on loop space*

arxiv:2009.00333*Fusion of implementers for spinors on the circle*

together with Peter Kristel

Adv. Math., to appear

arxiv:1905.00222*String geometry vs. spin geometry on loop spaces*

J. Geom. Phys. 97 (2015), 190-226

arxiv:1403.5656*Spin structures on loop spaces that characterize string manifolds*

Algebr. Geom. Topol. 16 (2016) 675–709

arxiv:1209.1731*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*String Connections and Chern-Simons Theory*

Trans. Amer. Math. Soc. 365 (2013), 4393-4432

arxiv:0906.0117

## Talks associated with this topic

*String Connections and Chern-Simons 2-Gerbes*

Workshop "Strings, Fields and Topology", Mathematisches Forschungsinstitut Oberwolfach, June 2009

Notes*String Connections and Supersymmetric Sigma Models*

Workshop "Homotopy theory and higher algebraic structures", University of California at Riverside, November 2009

Notes*Lectures on gerbes, loop spaces, and Chern-Simons theory*

Workshop "Chern-Simons Theory: Geometry, Topology and Physics", University of Pittsburgh, May 2013

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*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*Functorial Field Theories and Spin Geometry*

Mathematics Colloquium Series, New York University Abu Dhabi, November 2019

Notes*2-vector bundles, with applications to twisted K-theory and spin geometry*

Program "Higher Structures and Field Theory", Erwin-Schrödinger-Institut für Mathematische Physik, August 2020

Notes*String geometry*

Seminar "Renormalisation and Geometry", Universität Potsdam, June 2021

Video