Twisted loop transgression and higher Jandl gerbes over finite groupoids

TL;DR

This paper develops explicit constructions of twisted loop transgression maps for finite groupoids, linking Jandl gerbes to twisted vector bundles and relating to physical theories like orientifold string theory.

Contribution

It introduces explicit twisted loop transgression maps for finite groupoids and interprets their effects on categories of twisted vector bundles, connecting to physical theories.

Findings

Constructed explicit twisted loop transgression maps $ au_{ ext{pi}}$ and $ au_{ ext{pi}}^{ref}$.

Interpreted character theory and centers of twisted vector bundle categories in terms of flat sections.

Connected mathematical constructions to physical models like orientifold string and M-theory.

Abstract

Given a double cover $\pi: \mathcal{G} \rightarrow \hat{\mathcal{G}}$ 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 $\tau^{ref}_{\pi}(\hat{\lambda})$ 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.