Schreiber Cyclification of Orbifolds

An article that we have written:

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• Hisham Sati$\;$and$\;$ Urs Schreiber

  
  

  Cyclification of Orbifolds

  
  

  Comm. Math. Phys. 405 67 (2024)

  doi:10.1007/s00220-023-04929-w

  
  

  download:

  • pdf

  • arXiv:2212.13836

Abstract. Inertia orbifolds homotopy-quotiented by rotation of geometric loops play a fundamental role not only in ordinary cyclic cohomology, but more recently in constructions of equivariant quasi-elliptic cohomology and generally of transchromatic characters on generalized cohomology theories. Nevertheless, existing discussion of such cyclified stacks has been relying on ad-hoc component presentations with intransparent and unverified stacky homotopy type.

Following our previous formulation of transgression of cohomological charges (“double dimensional reduction”), we explain how cyclification of $\infty$-stacks is a fundamental and elementary base change construction over moduli stacks in cohesive higher topos theory (cohesive homotopy type theory). We prove that Ganter-Huan's extended inertia groupoid used to define equivariant quasi-elliptic cohomology is indeed a model for this intrinsically defined cyclification of orbifolds, and we show that cyclification implements transgression in group cohomology in general, and hence in particular the transgression of degree-4 twists of equivariant quasi-elliptic cohomology to degree-3 twists of orbifold K-theory on the cyclified orbifold.

As an application, we show that the universal shifted integral 4-class of equivariant 4-Cohomotopy theory on ADE-orbifolds induces the Platonic 4-twist of ADE-equivariant quasi-elliptic cohomology; and we close by explaining how this should relate to elliptic genera of the M5-brane, under our previously formulated Hypothesis H.

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