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
- A representation of the string 2-group
arxiv:2308.05139 - Lie 2-groups from loop group extensions
arxiv:2303.13176 - The stringor bundle
arxiv:2206.09797 - Connes fusion of spinors on loop space
arxiv:2012.08142 - Smooth Fock bundles, and spinor bundles on loop space
together with Peter Kristel
J. Diff. Geom., to appear
arxiv:2009.00333 - Fusion of implementers for spinors on the circle
together with Peter Kristel
Adv. Math. Vol. 402 (2022), 108325
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 - A representation of the string 2-group
TQFT Club Seminar, online, November 2022
Presentation Video - The stringor bundle
Workshop "Geometric/Topological Quantum Field Theories and Cobordisms", New York University Abu Dhabi, March 2023
Presentation - The stringor bundle
Conference "Geometries from Strings and Fields", Galileo Galilei Institute, July 2023
Video