Twisted loop transgression and higher Jandl gerbes over finite groupoids

Noohi, Behrang; Young, Matthew B

doi:10.2140/agt.2022.22.1663

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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

DOI: 10.2140/agt.2022.22.1663

Abstract

Given a double cover $π : 𝒢 \to \hat{𝒢}$ of finite groupoids, we explicitly construct cochain-level twisted loop transgression maps, $τ_{π}$ and $τ_{π}^{ref}$ , thereby associating to a Jandl $n$ –gerbe $\hat{λ}$ on $\hat{𝒢}$ a Jandl $(n - 1)$ –gerbe $τ_{π} (\hat{λ})$ on the quotient loop groupoid of $𝒢$ and an ordinary $(n - 1)$ –gerbe $τ_{π}^{ref} (\hat{λ})$ 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 $\hat{λ}$ –twisted $n$ –vector bundles over $\hat{𝒢}$ admits a natural interpretation in terms of flat sections of the $(n - 1)$ –vector bundle associated to $τ_{π}^{ref} (\hat{λ})$ (resp. the Real $(n - 1)$ –vector bundle associated to $τ_{π} (\hat{λ})$ ). 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.

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Keywords

groupoids, twisted equivariant $KR$–theory, Real representation theory

Mathematical Subject Classification 2010

Primary: 57R56

Secondary: 19L50, 20C30

References

Publication

Received: 1 December 2019

Revised: 9 March 2021

Accepted: 3 May 2021

Published: 10 October 2022

Authors

Behrang Noohi
School of Mathematical Sciences Queen Mary University of London London United Kingdom

Matthew B Young
Department of Mathematics and Statistics Utah State University Logan, UT United States

Volume 22, issue 4 (2022)