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Domenico Fiorenza, Hisham Sati, Urs Schreiber:
The character map in (twisted differential) non-abelian cohomology
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Abstract We extend the Chern character on K-theory, in its generalization to the Chern-Dold character on generalized cohomology theories, further 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, which we review in streamlined form, to a twisted non-abelian generalization of the de Rham theorem. We show that this 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 non-abelian cohomology, represented by higher bundles/higher gerbes (with higher connections). As a fundamental example we discuss the character map on twistorial Cohomotopy theory over 8-manifolds, which is a twisted non-abelian enhancement of the Chern-Dold character on 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.
Companion articles:
Domenico Fiorenza, Hisham Sati, Urs Schreiber:
Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation
Related talks:
Microscopic Brane Physics from Cohomotopy
talk at M-Theory and Mathematics
NYU Abu Dhabi, 2020
Twisted Cohomotopy implies M-theory anomaly cancellation
presentation at Strings2019
Brussels, 2019
Super p-Brane Theory emerging from Super Homotopy Theory
talk at StringMath2017,
Hamburg, 2017
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Last revised on February 23, 2021 at 08:11:12. See the history of this page for a list of all contributions to it.