nLab
curving

Contents

Idea

The differential 2-form-data in a circle 2-bundle with connection is sometimes called the “curving” of the connection data.

This terminology results from thinking of the circle 2-bundle equivalently as a bundle gerbe given by a transition-circle bundle on a suitable Cech groupoid. From this perspective the differential 1-form is an ordinary connection on a bundle for the transition bundles, and the 2-form data looks a bit like a variant of the curvature of these bundles, which not exactly being this curvature. Hence “curving”.

But of course the circle 2-bundle with connection has its own curvature differential 3-form. At least as soon as one passes to circle n-bundles with connection for arbitrary nn \in \mathbb{N}, the term “curving” becomes inappropriate.

Properties

(BU(1))(\mathbf{B}U(1))-principal 2-connections without curving

Circle 2-bundles “with connection but without curving” have as moduli 2-stack the delooping

B(BU(1) conn)\mathbf{B}(\mathbf{B}U(1)_{conn}) \in Smooth∞Grpd

of the moduli stack BU(1) conn\mathbf{B}U(1)_{conn} or circle bundles with connection.

The higher Atiyah groupoid of such a “pseudo-2-connection” 1:XB(BU(1) conn)\nabla_1 \colon X \to \mathbf{B}(\mathbf{B}U(1)_{conn}) is the corresponding higher Courant groupoid which Lie integrates the corresponding standard Courant Lie 2-algebroid. (See at higher Atiyah groupoid for details.)

Created on February 20, 2013 at 01:43:04. See the history of this page for a list of all contributions to it.