Multiplicative gerbes and Lie groups

Multiplicative gerbes are gerbes over Lie groups that are compatible with the group structure. They provide a geometric realization of the cohomology of the classifying space of the Lie group. Moreover, connections on multiplicative gerbes provide a geometrical realization of its differential refinement.

In my research I try to extend the general theory of multiplicative gerbes with connections, and I pursue essentially the following two applications. The first application is to Chern-Simons theories; these are 3-dimensional topological field theories of great importance in Mathematics and Physics. Multiplicative gerbes can help to understand Chern-Simons theories with very general gauge groups, in particular non-simply connected ones. In this context the gerbe generalizes the so-called "level" of the Chern-Simons theory.

The second application is about transgression to the loop group of the underlying Lie group. Under transgression, a multiplicative gerbe with connection becomes a central extension of the loop group. Multiplicative gerbes thus allow a finite-dimensional, higher-categorical perspective on the infinite-dimensional geometry of these central extensions.

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

Articles

  • Transgressive loop group extensions
    Math. Z. 286(1) 325-360, 2017
    arxiv:1502.05089  
  • 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  
  • Polyakov-Wiegmann Formula and Multiplicative Gerbes
    together with Krzysztof Gawedzki
    J. High Energy Phys. 09 (2009) 073
    arxiv:0908.1130  
  • Multiplicative Bundle Gerbes with Connection
    Differential Geom. Appl. 28(3), 313-340 (2010)
    arxiv:0804.4835  

Talks

  • Multiplicative Gerbes and Chern-Simons Theory
    Seminar "Topologie", Universität Bonn, October 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  
  • Transgressive central extensions of loop groups
    Conference "Colloquium on Algebras and Representations - Quantum 2016", Universidad Nacional de Córdoba, March 2016
    Notes