Leseprobe
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CHARACTER MAP IN NON-ABELIAN COHOMOLOGY, THE
Urs Schreiber, Domenico Fiorenza, Hisham Sati, et al.
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World Scientific Publishing Company 
Naturwissenschaften, Medizin, Informatik, Technik / Arithmetik, Algebra
Beschreibung
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, firstly, their rational approximation by generalized Chern character maps, and then, the resulting charge quantization laws in higher n-form gauge field theories appearing in string theory and the classification of topological quantum materials.
Although crucial for understanding famously elusive effects in strongly interacting physics, the relevant higher non-abelian cohomology theory ('higher gerbes') has had an esoteric reputation and remains underdeveloped.
Devoted to this end, this book's theme is that various generalized cohomology theories are best viewed through their classifying spaces (or moduli stacks) — not necessarily infinite-loop spaces — from which perspective the character map is really an incarnation of the fundamental theorem of rational homotopy theory, thereby not only uniformly subsuming the classical Chern character and a multitude of scattered variants that have been proposed, but now seamlessly applicable in the hitherto elusive generality of (twisted, differential, and) non-abelian cohomology.
In laying out this result with plenty of examples, this book provides a modernized introduction and review of fundamental classical topics: 1. abstract homotopy theory via model categories; 2. generalized cohomology in its homotopical incarnation; 3. rational homotopy theory seen via homotopy Lie theory, whose fundamental theorem we recast as a (twisted) non-abelian de Rham theorem, which naturally induces the (twisted) non-abelian character map.
Contents:
- Preface
- Introduction
- Non-abelian cohomology:
- Model Category Theory
- Non-abelian Cohomology Theories
- Twisted Non-abelian Cohomology
- Non-abelian de Rham Cohomology:
- Dgc-algebras and L∞-algebras
- ℝ-rational Homotopy Theory
- Non-abelian de Rham Theorem
- The (Differential) Non-abelian Character Map:
- Chern-Dold Character
- Chern-Weil Homomorphism
- Cheeger-Simons Homomorphism
- The Twisted (Differential) Non-abelian Character Map:
- Twisted Chern Character on Higher K-theory
- Twisted Differential Non-abelian Character
- Twisted Character on Twisted Differential Cohomotopy
- Bibliography
- Index
Readership: Graduate students, researchers in differential geometry, algebraic topology, and their applications to physics. Advanced undergraduate in mathematics and physics, novice researchers interested in a modern introduction to homotopy theory and techniques.
Key Features:
- It is the first systematic review of the Chern character map on K-theory and its variants, which are important tools in algebraic topology and differential geometry
- Provides, for the first time clearly, the mathematical basis for understanding one of the most important and profound topics of the past century: strongly-interacting and hence "non-perturbative" quantum systems
- Accessible exposition for both specialists and non-specialists, appended with background materials for students
- Concise but comprehensive introduction to the modern model category theory, not available elsewhere
- A theoretical work that comes with plenty of examples and draws many connections to its applications, not available elsewhere
