Schreiber The Character Map in Non-Abelian Cohomology

A book that we have written:

Cover blurb: This book presents a novel development of fundamental and fascinating aspects of algebraic topology and mathematical physics: “extra-ordinary” and further generalized cohomology theories enhanced to “twisted” and differential-geometric form with focus on their rational approximation by generalized Chern character maps and on the resulting charge quantization laws in higher n n -form gauge field theories appearing in string theory and in the classification of topological quantum materials.

Motivation for the conceptual re-development is the observation, laid out in the introductory chapter, that famous and famously elusive effects in strongly interacting (“non-perturbative”) physics demand “non-abelian” generalization of much of established generalized cohomology theory. But the relevant higher non-abelian cohomology theory (”higher gerbes“) has an esoteric reputation and has remained underdeveloped.

The book’s theme is that variously generalized cohomology theories are best viewed through their classifying spaces – possibly but not necessarily infinite-loop spaces – from which perspective the character map is really an incarnation of the fundamental theorem of rational homotopy theory, thereby uniformly subsuming not only the classical Chern character and a multitude of scattered variants that have been proposed, but now seamlessly applying in the previously elusive generality of (twisted, differential and) non-abelian cohomology.

In laying out this result with plenty of examples, we provide modernized introduction and review of fundamental classical topics: 1. abstract homotopy theory via model categories, 2. generalized cohomology in homotopical incarnation, 3. dg-algebraic rational homotopy theory, whose fundamental theorem we re-cast as a (twisted) non-abelian de Rham theorem which naturally induces the (twisted) non-abelian character map.

Technical 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:

Related talks:

Related articles (revolving around Hypothesis H):

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