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Register a new userConnections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing
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Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a new connective structure on the lifting gerbe associated to this problem. Our main result classifies all connections on the central extension of a given principal bundle. In particular, we find that admissible connections are in one-to-one correspondence with parallel trivializations of the lifting gerbe. Moreover, we prove a vanishing result for Neebs obstruction classes for finite-dimensional Lie groups.
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We introduce the notion of central extension of gerbes on a topological space. We then show that there are obstruction classes to lifting objects and isomorphisms in a central extension. We also discuss pronilpotent gerbes. These results are used in a subsequent paper to study twisted deformation quantization on algebraic varieties.
We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the role of a band, or structure 2-group. Upon choosing certain examples of Lie 2-groups, our axiomatic framework reproduces in a systematical way several known concepts of gerbes with connection: non-abelian differential cocycles, Breen-Messing gerbes, abelian and non-abelian bundle gerbes. These relationships convey a well-defined notion of surface holonomy from our axiomatic framework to each of these concrete models. Till now, holonomy was only known for abelian gerbes; our approach reproduces that known concept and extends it to non-abelian gerbes. Several new features of surface holonomy are exposed under its extension to non-abelian gerbes; for example, it carries an action of the mapping class group of the surface.
Let $(P, Y)$ be a bundle gerbe over a fibre bundle $Y to M$. We show that if $M$ is simply-connected and the fibres of $Y to M$ are connected and finite-dimensional then the Dixmier-Douady class of $(P, Y)$ is torsion. This corrects and extends an earlier result of the first author.
We introduce the notion of a general cup product bundle gerbe and use it to define the Weyl bundle gerbe on T x SU(n)/T. The Weyl map from T x SU(n)/T to SU(n) is then used to show that the pullback of the basic bundle gerbe on SU(n) defined by the second two authors is stably isomorphic to the Weyl bundle gerbe as SU(n)-equivariant bundle gerbes. Both bundle gerbes come equipped with connections and curvings and by considering the holonomy of these we show that these bundle gerbes are not D-stably isomorphic.
The aim of this paper is to introduce a cosymplectic analouge of conformal connection in a cosymplectic manifold and proved that if cosymplectic manifold M admits a cosymplectic conformal connection which is of zero curvature, then the Bochner curvature tensor of M vanishes.
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