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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mo class="MathClass-punc">:</mo> <mi mathvariant="bold-script">𝒢</mi><mo class="MathClass-rel">→</mo><mover accent="false"><mrow><mi mathvariant="bold-script">𝒢</mi></mrow><mo accent="true">^</mo></mover></math>$ of finite groupoids, we explicitly construct cochain-level twisted loop transgression maps, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>τ</mi></mrow><mrow><mi>π</mi></mrow></msub></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>π</mi></mrow><mrow><mi class="qopname">ref</mi><mo> ⁡ </mo></mrow></msubsup></math>$ , thereby associating to a Jandl $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>$ –gerbe $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="false"><mrow><mi>λ</mi></mrow><mo accent="true">^</mo></mover></math>$ on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="false"><mrow><mi mathvariant="bold-script">𝒢</mi></mrow><mo accent="true">^</mo></mover></math>$ a Jandl $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-open" stretchy="false">(</mo><mi>n</mi><mo class="MathClass-bin">−</mo><mn>1</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ –gerbe $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>τ</mi></mrow><mrow><mi>π</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mover accent="false"><mrow><mi>λ</mi></mrow><mo accent="true">^</mo></mover><mo class="MathClass-close" stretchy="false">)</mo></math>$ on the quotient loop groupoid of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">𝒢</mi></math>$ and an ordinary $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-open" stretchy="false">(</mo><mi>n</mi><mo class="MathClass-bin">−</mo><mn>1</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ –gerbe $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>π</mi></mrow><mrow><mi class="qopname">ref</mi><mo> ⁡ </mo></mrow></msubsup><mo class="MathClass-open" stretchy="false">(</mo><mover accent="false"><mrow><mi>λ</mi></mrow><mo accent="true">^</mo></mover><mo class="MathClass-close" stretchy="false">)</mo></math>$ on the unoriented quotient loop groupoid of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">𝒢</mi></math>$ . For $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi> <mo class="MathClass-rel">=</mo> <mn>1</mn><mo class="MathClass-punc">,</mo><mn>2</mn></math>$ , we prove that the character theory (resp. centre) of the category of Real $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="false"><mrow><mi>λ</mi></mrow><mo accent="true">^</mo></mover></math>$ –twisted $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>$ –vector bundles over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="false"><mrow><mi mathvariant="bold-script">𝒢</mi></mrow><mo accent="true">^</mo></mover></math>$ admits a natural interpretation in terms of flat sections of the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-open" stretchy="false">(</mo><mi>n</mi><mo class="MathClass-bin">−</mo><mn>1</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ –vector bundle associated to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>π</mi></mrow><mrow><mi class="qopname">ref</mi><mo> ⁡ </mo></mrow></msubsup><mo class="MathClass-open" stretchy="false">(</mo><mover accent="false"><mrow><mi>λ</mi></mrow><mo accent="true">^</mo></mover><mo class="MathClass-close" stretchy="false">)</mo></math>$ (resp. the Real $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-open" stretchy="false">(</mo><mi>n</mi><mo class="MathClass-bin">−</mo><mn>1</mn><mo class="MathClass-close" stretchy="false">)</mo></math>$ –vector bundle associated to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>τ</mi></mrow><mrow><mi>π</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mover accent="false"><mrow><mi>λ</mi></mrow><mo accent="true">^</mo></mover><mo class="MathClass-close" stretchy="false">)</mo></math>$ ). 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi></math>$ –theory.

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)