Basic equivariant gerbes on non-simply connected compact simple Lie groups

Krepski, Derek

doi:10.1016/j.geomphys.2018.06.016

Cited by 5 publications

(4 citation statements)

References 18 publications

(55 reference statements)

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“…, with respect to the conjugation action. For certain choices of inner product (parameterized by the so-called level ), there are a number of explicit constructions in the literature of a bundle gerbe with connective structure (G, B, γ) over G whose 3-curvature is η, as well as G-equivariant versions-e.g., see [25], [14] and [21]. ⋄ 3.2.…”

Section: 1mentioning

confidence: 99%

Multiplicative vector fields on bundle gerbes

Krepski,

Vaughan

2020

Preprint

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This paper makes use of multiplicative vector fields on Lie groupoids to model infinitesimal symmetries of S 1 -bundle gerbes. It is shown that a connective structure on a bundle gerbe gives rise to a natural horizontal lift of multiplicative vector fields to the bundle gerbe, and that the 3-curvature presents the obstruction to the horizontal lift being a morphism of Lie 2-algebras. Connectionpreserving multiplicative vector fields on a bundle gerbe with connective structure are shown to inherit a natural Lie 2-algebra structure; moreover, this Lie 2-algebra is quasi-isomorphic to the Poisson-Lie 2-algebra of the 2-plectic base manifold (M, χ), where χ is the 3-curvature of the connective structure. As an application of this result, we give an analogue of a formula of Kostant in the 2-plectic and quasi-Hamiltonian context.

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Section: 1mentioning

confidence: 99%

Multiplicative vector fields on bundle gerbes

Krepski,

Vaughan

2020

Preprint

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“…For with , as , there is only one nontrivial torsion class and this can be realised by a lifting gerbe, following [12, Remark 5.1]. Namely, there is a nontrivial central extension where the underlying manifold of is with multiplication arising from the nontrivial 2-cocycle with [8].…”

Section: Examples and Applicationsmentioning

confidence: 99%

“…For G = PSO(4n) with n > 1, as H 3 (PSO(4n), Z) ≃ Z ⊕ Z/2, there is only one nontrivial torsion class and this can be realised by a lifting gerbe, following [12,Remark 5.1]. Namely, there is a nontrivial central extension…”

Section: Example 43 Consider the Infinite Direct Summentioning

confidence: 99%

Many Finite-Dimensional Lifting Bundle Gerbes Are Torsion

Roberts

2021

Bull. Aust. Math. Soc.

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Many bundle gerbes are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson [‘A note on bundle gerbes and infinite-dimensionality’, J. Aust. Math. Soc.90(1) (2011), 81–92] proved that gerbes on simply-connected manifolds, built from finite-dimensional fibre bundles with connected fibres, always have a torsion $DD$ -class. I prove an analogous result for a wide class of gerbes built from principal bundles, relaxing the requirements on the fundamental group of the base and the connected components of the fibre, allowing both to be nontrivial. This has consequences for possible models for basic gerbes, the classification of crossed modules of finite-dimensional Lie groups, the coefficient Lie-2-algebras for higher gauge theory on principal 2-bundles and finite-dimensional twists of topological K-theory.

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“…This construction is implicit in Meinrenken's construction [6] and appears explicitly in [30]. In the second step, descent to a non-simply-connected quotient G := G/Z, where Z ⊂ Z(G), is performed along G/ /G → G/ / G. The required Z-equivariant structures have been provided Lie-theoretically by Gawȩdzki-Reis [31] (without the G-equivariance) and recently -including the G-equivariance -by Krepski [32].…”

Section: Gerbes Over Lie Groupoidsmentioning

confidence: 99%