Sigma Models

Sigma models are two-dimensional classical conformal field theories in which the fields are maps from a Riemann surface to a fixed Riemannian manifold, the target space. These field theories typically suffer from an anomaly that can be cancelled by a so-called topological term. This term is exactly the holonomy of a gerbe-connection over the targetspace.

My work in this area deals with defining the topological term in situations where additional structures are either added or omitted. For example, omitting the orientation of the surface requires a so-called Jandl structure on the Gerbe.

An important special case are Lie groups as target spaces, and the corresponding sigma models are called Wess-Zumino-Witten models. Here, one aspect of my work is to give a Lie-theoretical classification of the relevant additional structures.

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

Articles associated with this topic

  • Geometric T-duality: Buscher rules in general topology
    Ann. Henri Poincaré, to appear
  • Smooth Functorial Field Theories from B-Fields and D-Branes
    together with Severin Bunk
    J. Homotopy Relat. Struct. 16.1 (2021): 75-153
  • Transgression of D-branes
    together with Severin Bunk
    Adv. Theor. Math. Phys., Vol. 25, No. 5 (2021), pp. 1095-1198.
  • The gauging of two-dimensional bosonic sigma models on world-sheets with defects
    together with Rafal R. Suszek, Krzysztof Gawedzki
    Rev. Math. Phys 25 (2013) 1350010
  • Global Gauge Anomalies in two-dimensional Bosonic Sigma Models
    together with Rafal R. Suszek, Krzysztof Gawedzki
    Commun. Math. Phys. 302 (2), 513-580 (2011)
  • Bundle Gerbes and Surface Holonomy
    together with Christoph Schweigert, Thomas Nikolaus, Jürgen Fuchs
    Proceedings of the 5th European Congress of Mathematics, edited by A. Ran, H. te Riele and J. Wiegerinck, EMS Publishing House, 2008, 167-197
  • Bundle Gerbes for Orientifold Sigma Models
    together with Krzysztof Gawedzki, Rafal R. Suszek
    Adv. Theor. Math. Phys. 15 (3), 621-688 (2011)
  • Bi-branes: Target Space Geometry for World Sheet topological Defects
    together with Christoph Schweigert, Jürgen Fuchs
    J. Geom. Phys. 58(5), 576-598 (2008)
  • WZW Orientifolds and finite Group Cohomology
    together with Krzysztof Gawedzki, Rafal R. Suszek
    Commun. Math. Phys. 284(1), 1–49 (2007)
  • Unoriented WZW Models and Holonomy of Bundle Gerbes
    together with Christoph Schweigert, Urs Schreiber
    Commun. Math. Phys. 274(1), 31–64 (2007)

Talks associated with this topic

  • Introduction to Gerbes in Conformal Field Theory
    Workshop "Gerbes, twisted K-theory and conformal field theory", Mathematisches Forschungsinstitut Oberwolfach, August 2005
  • Gerbes in unoriented WZW-Models
    Summer school "Modern Mathematical Physics IV", University of Belgrade, September 2006
  • Geometry for 2-Form Gauge Fields
    Conference "Tagung des Sonderforschungsbereichs Particles, Strings and the Early Universe", Deutsches Elektronen-Synchrotron DESY Zeuthen, February 2008
  • String Connections and Supersymmetric Sigma Models
    Workshop "Homotopy theory and higher algebraic structures", University of California at Riverside, November 2009
  • String structures and supersymmetric sigma models
    Program "Higher structures in string theory and quantum field theory", Erwin-Schrödinger-Institut für Mathematische Physik, December 2015
  • An introduction to higher parallel transport
    Seminar "Algebraic and combinatorial perspectives in the mathematical sciences", online, September 2020
    Presentation   Video  
  • Geometric T-Duality: Buscher rules in general topology
    Program "Higher Structures and Field Theory", Erwin-Schrödinger-Institut für Mathematische Physik, August 2022