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.
- 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
- 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