Twisted loop transgression and higher Jandl gerbes over finite groupoids

Abstract

Given a double cover $\pi :\mathcal{G}\to \hat{\mathcal{G}}$ of finite groupoids, we explicitly construct twisted loop transgression maps, ${\tau }_{\pi }$ and ${\tau }_{\pi }^{ref}$, thereby associating to a Jandl $n$-gerbe $\hat{\lambda }$ on $\hat{\mathcal{G}}$ a Jandl $(n-1)$-gerbe ${\tau }_{\pi }(\hat{\lambda })$ on the quotient loop groupoid of $\mathcal{G}$ and an ordinary $(n-1)$-gerbe ${\tau }_{\pi }^{ref}(\hat{\lambda })$ on the unoriented quotient loop groupoid of $\mathcal{G}$. For $n=1,2$, we interpret the character theory (resp. centre) of the category of Real $\hat{\lambda }$-twisted $n$-vector bundles over $\hat{\mathcal{G}}$ in terms of flat sections of the $(n-1)$-vector bundle associated to ${\tau }_{\pi }^{ref}(\hat{\lambda })$ (resp. the Real $(n-1)$-vector bundle associated to ${\tau }_{\pi }(\hat{\lambda })$). We relate our results to Real versions of twisted Drinfeld doubles and pointed fusion categories and to discrete torsion in orientifold string and M-theory.

arXiv comment33 pages

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Cited by in corpus (1)

• Real Representations of $C_2$-Graded Groups: The Antilinear Theory