Differential Geometry of Gerbes
Abstract
We define in a global manner the notion of a connective structure for a gerbe on a space X. When the gerbe is endowed with trivializing data with respect to an open cover of X, we describe this connective structure in two separate ways, which extend from abelian to general gerbes the corresponding descriptions due to J. L. Brylinski and N. Hitchin. We give a global definition of the 3curvature of this connective structure as a 3form on X with values in the Lie stack of the gauge stack of the gerbe. We also study this notion locally in terms of more traditional Lie algebravalued 3forms. The Bianchi identity, which the curvature of a connection on a principal bundle satisfies, is replaced here by a more elaborate equation.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2001
 DOI:
 10.48550/arXiv.math/0106083
 arXiv:
 arXiv:math/0106083
 Bibcode:
 2001math......6083B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Category Theory;
 Mathematics  Differential Geometry;
 High Energy Physics  Theory
 EPrint:
 78 pages. Uses XyPic. A number of significant additions have been incorporated into this version. This includes a description of the coboundary relations in full generality, as well as a Cechde Rham interpretation of the cocycle and coboundary relations for the 3curvature of a gerbe with connection. New and more conceptual proofs, which make use of bitorsor diagrams, are given for most of these relations