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Twisted loop transgression and higher Jandl gerbes over finite groupoids

Behrang Noohi and Matthew B Young

Algebraic & Geometric Topology 22 (2022) 1663–1712

Given a double cover π: 𝒢𝒢^ of finite groupoids, we explicitly construct cochain-level twisted loop transgression maps, τπ and τπref , thereby associating to a Jandl n–gerbe λ^ on 𝒢^ a Jandl (n1)–gerbe τπ(λ^) on the quotient loop groupoid of 𝒢 and an ordinary (n1)–gerbe τπref (λ^) on the unoriented quotient loop groupoid of 𝒢. For n = 1,2, we prove that the character theory (resp. centre) of the category of Real λ^–twisted n–vector bundles over 𝒢^ admits a natural interpretation in terms of flat sections of the (n1)–vector bundle associated to τπref (λ^) (resp. the Real (n1)–vector bundle associated to τπ(λ^)). We relate our results to Real versions of twisted Drinfeld doubles of finite groups and fusion categories and to discrete torsion in orientifold string theory and M–theory.

groupoids, twisted equivariant $KR$–theory, Real representation theory
Mathematical Subject Classification 2010
Primary: 57R56
Secondary: 19L50, 20C30
Received: 1 December 2019
Revised: 9 March 2021
Accepted: 3 May 2021
Published: 10 October 2022
Behrang Noohi
School of Mathematical Sciences
Queen Mary University of London
United Kingdom
Matthew B Young
Department of Mathematics and Statistics
Utah State University
Logan, UT
United States