Principal 2-groupoid bundles with adjusted connections
Abstract
Description
We develop a radically new theory of higher principal bundles with adjusted connections,
avoiding the fake flatness condition. The theory can be realised in any model of higher Lie
groupoids, such as higher (geometric) stacks. We will, particularly, focus on Lie 2-groupoids
to provide examples of our theory. In this setup, higher principal bundles with adjusted
connections are principal bundles without connections in the category of NQ-manifolds.
Particularly, higher principal G-bundles with flat connections over X are given by morphisms T[1]X × Pair(Θ) → G, where Pair(Θ) is the pair groupoid of the shifted real line.
In particular, this lifts the Ševera differentiation of higher Lie groupoids to NQ-manifolds
to a categorically more appealing definition A (G) as representation of this higher stack.
Higher principal G-bundles with flat connections over a manifold X are higher principal
A (G) bundles (without connection) over T[1]X, the shifted tangent bundle. To lift flatness,
adjustment groupoids Aκw
(G) are introduced. Adjustment groupoids are higher NQ-Lie
groupoids satisfying some axioms. These axioms are designed so that our expectations of
higher principal bundles with connections are fulfilled.
We develop a theory of 2-geometric 2-stacks over the site of NQ-manifolds and smooth
manifolds, together with various adjunctions between them. So, we can realise this abstract
theory in this model.
For Lie groupoids, we prove that adjustment groupoids are classified by Cartan connections. Furthermore, we show that our theory subsumes the existing notion of adjustments for
strict Lie 2-groups.
We develop a local theory by defining the adjusted BRST-presheaf. Extra structure
is needed in this definition to systematically eliminate the unwanted infinitesimal gauge
transformations associated with the curvature that appears in the naive AKSZ construction.
In all our examples, the structure that the adjustment groupoids carry can be linearised to
adjustments for the local theory.