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
String structures and loop spaces
Encyclopedia of Mathematical Physics (2nd edition), to appear
arxiv:2312.12998A representation of the string 2-group
arxiv:2308.05139Lie 2-groups from loop group extensions
arxiv:2303.13176The stringor bundle
arxiv:2206.09797Connes fusion of spinors on loop space
together with Peter Kristel
Compos. Math., to appear
arxiv:2012.08142Smooth Fock bundles, and spinor bundles on loop space
together with Peter Kristel
J. Diff. Geom., to appear
arxiv:2009.00333Fusion of implementers for spinors on the circle
together with Peter Kristel
Adv. Math. Vol. 402 (2022), 108325
arxiv:1905.00222String geometry vs. spin geometry on loop spaces
J. Geom. Phys. 97 (2015), 190-226
arxiv:1403.5656Spin structures on loop spaces that characterize string manifolds
Algebr. Geom. Topol. 16 (2016) 675–709
arxiv:1209.1731A 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:1201.5052Lifting Problems and Transgression for Non-Abelian Gerbes
together with Thomas Nikolaus
Adv. Math. 242 (2013) 50-79
arxiv:1112.4702String 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
NotesString Connections and Supersymmetric Sigma Models
Workshop "Homotopy theory and higher algebraic structures", University of California at Riverside, November 2009
NotesLectures on gerbes, loop spaces, and Chern-Simons theory
Workshop "Chern-Simons Theory: Geometry, Topology and Physics", University of Pittsburgh, May 2013
NotesString structures and supersymmetric sigma models
Program "Higher structures in string theory and quantum field theory", Erwin-Schrödinger-Institut für Mathematische Physik, December 2015
NotesString geometry and spin geometry on loop spaces
Parallel session "Mathematical aspects of string theory and string geometry", Friedrich-Schiller-Universität Jena, July 2016
NotesString connections and loop spaces
Workshop "Loop spaces, supersymmetry and index theory", Nankai University at Tianjin, July 2017
PresentationFusion in loop spaces
Workshop "Geometric Quantization", Banff International Research Station for Mathematical Innovation and Discovery, April 2018
VideoFunctorial Field Theories and Spin Geometry
Mathematics Colloquium Series, New York University Abu Dhabi, November 2019
Notes2-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
NotesString geometry
Seminar "Renormalisation and Geometry", Universität Potsdam, June 2021
VideoA representation of the string 2-group
TQFT Club Seminar, online, November 2022
Presentation VideoThe stringor bundle
Workshop "Geometric/Topological Quantum Field Theories and Cobordisms", New York University Abu Dhabi, March 2023
PresentationThe stringor bundle
Conference "Geometries from Strings and Fields", Galileo Galilei Institute, July 2023
Video