TY - JOUR
T1 - G-gerbes, principal 2-group bundles and characteristic classes
AU - Ginot, Grégory
AU - Stiénon, Mathieu
N1 - Publisher Copyright:
© 2015, International Press of Boston, Inc. All rights reserved.
PY - 2015
Y1 - 2015
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.
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U2 - 10.4310/JSG.2015.v13.n4.a6
DO - 10.4310/JSG.2015.v13.n4.a6
M3 - Article
AN - SCOPUS:84962624842
SN - 1527-5256
VL - 13
SP - 1001
EP - 1048
JO - Journal of Symplectic Geometry
JF - Journal of Symplectic Geometry
IS - 4
ER -