On the Chern character in Higher Twisted K-theory and spherical T-duality

Macdonald, Lachlan; Mathai, Varghese; Saratchandran, Hemanth

doi:10.48550/arxiv.2007.02507

Cited by 1 publication

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“…(iii) A map of this form has been defined in [MMS20, §2.2] by direct construction on form representatives. However, [MMS20,Thm. 4.19] implies that this component construction coincides with the rationalization map (Def.…”

Section: Cheeger-simons Homomorphismmentioning

confidence: 99%

The character map in (twisted differential) non-abelian cohomology

Fiorenza,

Sati,

Schreiber

2020

Preprint

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The Chern character on K-theory has a natural extension to arbitrary generalized cohomology theories known as the Chern-Dold character. Here we further extend this to (twisted, differential) non-abelian cohomology theories, where its target is a non-abelian de Rham cohomology of twisted L ∞ -algebra valued differential forms. The construction amounts to leveraging the fundamental theorem of dg-algebraic rational homotopy theory to a twisted non-abelian generalization of the de Rham theorem. We show that the non-abelian character reproduces, besides the Chern-Dold character, also the Chern-Weil homomorphism as well as its secondary Cheeger-Simons homomorphism on (differential) non-abelian cohomology in degree 1, represented by principal bundles (with connection); and thus generalizes all these to higher (twisted, differential) non-abelian cohomology, represented by higher bundles/higher gerbes (with higher connections). As a fundamental example, we discuss the twisted non-abelian character map on twistorial Cohomotopy theory over 8-manifolds, which can be viewed as a twisted non-abelian enhancement of topological modular forms (tmf) in degree 4. This turns out to exhibit a list of subtle topological relations that in high energy physics are thought to govern the charge quantization of fluxes in M-theory.

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Section: Cheeger-simons Homomorphismmentioning

confidence: 99%