nLab basic gerbe

Idea

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

For that to make verbatim sense we need for instance that $H^3(G;\mathbb{Z}) \,\simeq\, \mathbb{Z}$, which is the case notably for simply connected compact simple Lie groups $G$, such as SU(n) for $n \geq 2$ or Spin(n) for $n \geq 3$ (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 $H^3(G;\mathbb{R})$ by a differential 3-form which is $G$-bi-invariant (i.e. invariant under pullback of differential forms along the action of $G$ on itself by multiplication.)

With due care, the basic bundle gerbe itself has a multiplicative structure covering that on $G$ 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).

References

The original articles:

• Krzysztof Gawędzki, Nuno Reis, WZW branes and gerbes, Rev. Math. Phys. 14 (2002) 1281-1334 [arXiv:hep-th/0205233, doi:10.1142/S0129055X02001557]

• Eckhard Meinrenken, The basic gerbe over a compact simple Lie group, Enseign. Math. (2) 49 (2003), no. 3-4, 307-333 [arXiv:math/0209194, e-periodica]

• Krzysztof Gawędzki, Nuno Reis, Basic gerbe over non-simply connected compact groups, Journal of Geometry and Physics 50 1-4 (2004) 28–55 [arXiv:0307010, doi:10.1016/j.geomphys.2003.11.004]

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

• Michael K. Murray, Danny Stevenson, The basic bundle gerbe on unitary groups, Journal of Geometry and Physics 58 11 (2008) 1571-1590 [doi:10.1016/j.geomphys.2008.07.006, arXiv:0804.3464]

Review:

• Christoph Schweigert, Konrad Waldorf, Gerbes and Lie Groups, in: Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288 (2011) [arXiv:0710.5467, doi:10.1007/978-0-8176-4741-4_10]

As a model for the string 2-group:

• Konrad Waldorf, A Construction of String 2-Group Models using a Transgression-Regression Technique, Contemp. Math. 584 (2012) 99-115 [arXiv:1201.5052, doi:10.1090/conm/584/11588]

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, §3.4.2 in: A higher stacky perspective on Chern-Simons theory, in: Mathematical Aspects of Quantum Field Theories, Mathematical Physics Studies, Springer (2015) 153-211 [arXiv:1301.2580, doi:10.1007/978-3-319-09949-1_6]

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, §4.1 in: Higher $U(1)$-gerbe connections in geometric prequantization, Rev. Math. Phys. 28 06 1650012 (2016) [arXiv:1304.0236, doi:10.1142/S0129055X16500124]