nLab basic gerbe



With some abuse of terminology, by the “basic gerbe” over a Lie group GG authors mean, following Meinrenken 2003, a U(1)U(1)-bundle gerbe (typically understood with 2-connection) on the underlying topological space (or underlying smooth manifold) of GG, whose Dixmier-Douady class is a generator (whence “basic”) of the ordinary cohomology-group H 3(G;)H^3(G;\mathbb{Z}).

For that to make verbatim sense we need for instance that H 3(G;)H^3(G;\mathbb{Z}) \,\simeq\, \mathbb{Z}, which is the case notably for simply connected compact simple Lie groups GG, such as SU(n) for n2n \geq 2 or Spin(n) for n3n \geq 3 (but the notion has been considered more generally).

If this bundle gerbe is considered with 2-connection, then the curvature 3-form is typically understood to be a de Rham representative of H 3(G;)H^3(G;\mathbb{R}) by a differential 3-form which is GG-bi-invariant (i.e. invariant under pullback of differential forms along the action of GG on itself by multiplication.)

With due care, the basic bundle gerbe itself has a multiplicative structure covering that on GG and making it a 2-group, as such equivalent to the string 2-group (Waldorf 2012).

With bundle gerbes understood as a Kalb-Ramond background field for a string (its background B-field), the basic gerbes define the worldsheet sigma-model known as the Wess-Zumino-Witten model, which was the original motivation of Gawędzki & Reis 2002. In terms of higher prequantum geometry the bundle gerbe is the prequantum circle 2-bundle of the theory (Fiorenza, Sati & Schreiber 2015, Fiorenza, Rogers & Schreiber 2016).


The original articles:

A specific construction for unitary groups and generalization to groups with unitary representations on a Hilbert space:


As a model for the string 2-group:

and as the prequantum circle 2-bundle of the WZW model discussed in smooth infinity-groupoids:

Last revised on November 24, 2023 at 08:31:13. See the history of this page for a list of all contributions to it.