String geometry

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

  • A representation of the string 2-group

  • Lie 2-groups from loop group extensions

  • The stringor bundle

  • Connes fusion of spinors on loop space
    together with Peter Kristel
    Compos. Math., to appear

  • Smooth Fock bundles, and spinor bundles on loop space
    together with Peter Kristel
    J. Diff. Geom., to appear

  • Fusion of implementers for spinors on the circle
    together with Peter Kristel
    Adv. Math. Vol. 402 (2022), 108325

  • String geometry vs. spin geometry on loop spaces
    J. Geom. Phys. 97 (2015), 190-226

  • Spin structures on loop spaces that characterize string manifolds
    Algebr. Geom. Topol. 16 (2016) 675–709

  • 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

  • String Connections and Chern-Simons Theory
    Trans. Amer. Math. Soc. 365 (2013), 4393-4432

Talks associated with this topic

  • String Connections and Chern-Simons 2-Gerbes
    Workshop "Strings, Fields and Topology", Mathematisches Forschungsinstitut Oberwolfach, June 2009

  • String Connections and Supersymmetric Sigma Models
    Workshop "Homotopy theory and higher algebraic structures", University of California at Riverside, November 2009

  • Lectures on gerbes, loop spaces, and Chern-Simons theory
    Workshop "Chern-Simons Theory: Geometry, Topology and Physics", University of Pittsburgh, May 2013

  • String structures and supersymmetric sigma models
    Program "Higher structures in string theory and quantum field theory", Erwin-Schrödinger-Institut für Mathematische Physik, December 2015

  • 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

  • Functorial Field Theories and Spin Geometry
    Mathematics Colloquium Series, New York University Abu Dhabi, November 2019

  • 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

  • String geometry
    Seminar "Renormalisation and Geometry", Universität Potsdam, June 2021

  • 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

  • The stringor bundle
    Conference "Geometries from Strings and Fields", Galileo Galilei Institute, July 2023