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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
##### Abstract

Given a double cover $\pi :\mathsc{𝒢}\to \stackrel{^}{\mathsc{𝒢}}$ of finite groupoids, we explicitly construct cochain-level twisted loop transgression maps, ${\tau }_{\pi }$ and ${\tau }_{\pi }^{\mathrm{ref}}$, thereby associating to a Jandl $n$–gerbe $\stackrel{^}{\lambda }$ on $\stackrel{^}{\mathsc{𝒢}}$ a Jandl $\left(n-1\right)$–gerbe ${\tau }_{\pi }\left(\stackrel{^}{\lambda }\right)$ on the quotient loop groupoid of $\mathsc{𝒢}$ and an ordinary $\left(n-1\right)$–gerbe ${\tau }_{\pi }^{\mathrm{ref}}\left(\stackrel{^}{\lambda }\right)$ on the unoriented quotient loop groupoid of $\mathsc{𝒢}$. For $n=1,2$, we prove that the character theory (resp. centre) of the category of Real $\stackrel{^}{\lambda }$–twisted $n$–vector bundles over $\stackrel{^}{\mathsc{𝒢}}$ admits a natural interpretation in terms of flat sections of the $\left(n-1\right)$–vector bundle associated to ${\tau }_{\pi }^{\mathrm{ref}}\left(\stackrel{^}{\lambda }\right)$ (resp. the Real $\left(n-1\right)$–vector bundle associated to ${\tau }_{\pi }\left(\stackrel{^}{\lambda }\right)$). 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.

##### Keywords
groupoids, twisted equivariant $KR$–theory, Real representation theory
##### Mathematical Subject Classification 2010
Primary: 57R56
Secondary: 19L50, 20C30