G-gerbes, principal 2-group bundles and characteristic classes

Grégory Ginot, Mathieu Stiénon

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let G be a Lie group and G → Aut(G) be the canonical group homomorphism induced by the adjoint action of a group on itself. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group [G → Aut(G)]-bundles over Lie groupoids and, on the other hand, G-extensions of Lie groupoids (i.e. between principal [G→Aut(G)]- bundles over dierentiable stacks and G-gerbes over dierentiable stacks). This approach also allows us to identify G-bound gerbes and [Z(G) → 1]-group bundles over dierentiable stacks, where Z(G) is the center of G. We also introduce universal characteristic classes for 2-group bundles. For groupoid central G-extensions, we introduce Dixmier-Douady classes that can be computed from connection-type data generalizing the ones for bundle gerbes. We prove that these classes coincide with universal characteristic classes. As a corollary, we obtain further that Dixmier-Douady classes are integral.

Original languageEnglish (US)
Pages (from-to)1001-1048
Number of pages48
JournalJournal of Symplectic Geometry
Volume13
Issue number4
DOIs
StatePublished - Jan 1 2015

Fingerprint

Gerbes
Characteristic Classes
Bundle
Groupoids
Morita Equivalence
Groupoid
One to one correspondence
Equivalence class
Homomorphism
Corollary
Class

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Cite this

@article{43964083485245caad113a7b7d85c71a,
title = "G-gerbes, principal 2-group bundles and characteristic classes",
abstract = "Let G be a Lie group and G → Aut(G) be the canonical group homomorphism induced by the adjoint action of a group on itself. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group [G → Aut(G)]-bundles over Lie groupoids and, on the other hand, G-extensions of Lie groupoids (i.e. between principal [G→Aut(G)]- bundles over dierentiable stacks and G-gerbes over dierentiable stacks). This approach also allows us to identify G-bound gerbes and [Z(G) → 1]-group bundles over dierentiable stacks, where Z(G) is the center of G. We also introduce universal characteristic classes for 2-group bundles. For groupoid central G-extensions, we introduce Dixmier-Douady classes that can be computed from connection-type data generalizing the ones for bundle gerbes. We prove that these classes coincide with universal characteristic classes. As a corollary, we obtain further that Dixmier-Douady classes are integral.",
author = "Gr{\'e}gory Ginot and Mathieu Sti{\'e}non",
year = "2015",
month = "1",
day = "1",
doi = "10.4310/JSG.2015.v13.n4.a6",
language = "English (US)",
volume = "13",
pages = "1001--1048",
journal = "Journal of Symplectic Geometry",
issn = "1527-5256",
publisher = "International Press of Boston, Inc.",
number = "4",

}

G-gerbes, principal 2-group bundles and characteristic classes. / Ginot, Grégory; Stiénon, Mathieu.

In: Journal of Symplectic Geometry, Vol. 13, No. 4, 01.01.2015, p. 1001-1048.

Research output: Contribution to journalArticle

TY - JOUR

T1 - G-gerbes, principal 2-group bundles and characteristic classes

AU - Ginot, Grégory

AU - Stiénon, Mathieu

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Let G be a Lie group and G → Aut(G) be the canonical group homomorphism induced by the adjoint action of a group on itself. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group [G → Aut(G)]-bundles over Lie groupoids and, on the other hand, G-extensions of Lie groupoids (i.e. between principal [G→Aut(G)]- bundles over dierentiable stacks and G-gerbes over dierentiable stacks). This approach also allows us to identify G-bound gerbes and [Z(G) → 1]-group bundles over dierentiable stacks, where Z(G) is the center of G. We also introduce universal characteristic classes for 2-group bundles. For groupoid central G-extensions, we introduce Dixmier-Douady classes that can be computed from connection-type data generalizing the ones for bundle gerbes. We prove that these classes coincide with universal characteristic classes. As a corollary, we obtain further that Dixmier-Douady classes are integral.

AB - Let G be a Lie group and G → Aut(G) be the canonical group homomorphism induced by the adjoint action of a group on itself. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group [G → Aut(G)]-bundles over Lie groupoids and, on the other hand, G-extensions of Lie groupoids (i.e. between principal [G→Aut(G)]- bundles over dierentiable stacks and G-gerbes over dierentiable stacks). This approach also allows us to identify G-bound gerbes and [Z(G) → 1]-group bundles over dierentiable stacks, where Z(G) is the center of G. We also introduce universal characteristic classes for 2-group bundles. For groupoid central G-extensions, we introduce Dixmier-Douady classes that can be computed from connection-type data generalizing the ones for bundle gerbes. We prove that these classes coincide with universal characteristic classes. As a corollary, we obtain further that Dixmier-Douady classes are integral.

UR - http://www.scopus.com/inward/record.url?scp=84962624842&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84962624842&partnerID=8YFLogxK

U2 - 10.4310/JSG.2015.v13.n4.a6

DO - 10.4310/JSG.2015.v13.n4.a6

M3 - Article

AN - SCOPUS:84962624842

VL - 13

SP - 1001

EP - 1048

JO - Journal of Symplectic Geometry

JF - Journal of Symplectic Geometry

SN - 1527-5256

IS - 4

ER -