The basic gerbe over a compact simple Lie group. (English) Zbl 1061.53034

Meinrenken, Eckhard

Enseign. Math., II. Sér. 49, No. 3-4, 307-333 (2003).

In [Loop spaces, characteristic classes and geometric quantization (Birkhäuser, Boston) (1993; Zbl 0823.55002)], J.-L. Brylinski gave a general construction of a gerbe with connection, for any integral closed 3-form on any 2-connected manifold. In the case of a compact, simply connected simple Lie group \(G\) one looks for an explicit, finite dimensional description of an equivariant gerbe over \(G\), whose equivariant 3-curvature represents the class of the generator of \(H_G^3(G;{\mathbb Z})\). In [Rev. Math. Phys. 14, 1281–1334 (2002; Zbl 1033.81067)] this problem was solved by K. Gawȩdzki and N. Reis for \(\text{ SU}(n)\) and non-simply connected groups covered by \(\text{ SU}(n)\), in terms of transition line bundles due to D. Chatterjee and N. Hitchin.
In the paper under review, the author solves the problem for any compact, simply connected simple Lie group. He uses the bundle gerbes introduced by M. K. Murray [J. Lond. Math. Soc. 54, 403–416 (1996; Zbl 0867.55019)] and develops a gluing construction for equivariant bundles gerbes. As mentioned in the introduction, another construction has been given by K. Behrend and P. Xu and B. Zhang in [C. R. Math. Acad. Sci. Paris 336, 251–256 (2003; Zbl 1068.58010)]. Quote also an extension of the Meinrenken’s construction to the non simply connected case by K. Gawȩdzki and N. Reis in [“Basic gerbe over non-simply connected compact groups”, J. Geom. Phys. 50, 28–55 (2004)].

Reviewer: Daniel Tanré (Villeneuve d’ Ascq)

MSC:

53C29	Issues of holonomy in differential geometry
55R65	Generalizations of fiber spaces and bundles in algebraic topology
53C30	Differential geometry of homogeneous manifolds

Keywords:

gerbe; curvature; simple Lie group