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.
- Transgressive loop group extensions
Math. Z. 286(1) 325-360, 2017
- 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
- Polyakov-Wiegmann Formula and Multiplicative Gerbes
together with Krzysztof Gawedzki
J. High Energy Phys. 09 (2009) 073
- Multiplicative Bundle Gerbes with Connection
Differential Geom. Appl. 28(3), 313-340 (2010)
- Multiplicative Gerbes and Chern-Simons Theory
Seminar "Topologie", Universität Bonn, October 2009
- Lectures on gerbes, loop spaces, and Chern-Simons theory
Workshop "Chern-Simons Theory: Geometry, Topology and Physics", University of Pittsburgh, May 2013
- Transgressive central extensions of loop groups
Conference "Colloquium on Algebras and Representations - Quantum 2016", Universidad Nacional de Córdoba, March 2016