Geometric Orbifold Cohomology

1. Auflage 2026
ISBN: 978-1-041-14751-0
Verlag: Taylor & Francis Ltd

Buch, Englisch, 344 Seiten, Format (B × H): 156 mm x 234 mm

Topological phases of quantum materials and brane charges in M-theory are measured by extraordinary cohomology theories defined on orbifold spacetimes. Geometric Orbifold Cohomology presents a modernized and enhanced formulation of these theories, establishing a rigorous framework for nonabelian and differential cohomology in the setting of higher geometry. Motivated by cutting-edge problems in mathematical physics—specifically the analysis of M-brane charges and topological insulators—this monograph bridges the gap between classical equivariant topology and modern cohesive homotopy theory. It fills a critical gap in the literature by offering a unified perspective where orbifold geometry is treated synthetically via modal operators in higher topos theory, allowing for a precise treatment of geometric singularities and differential structures.

• Accessible Foundations: Begins with a streamlined reconstruction of twisted nonabelian orbifold cohomology and orbifold K-theory using the language of topological groupoids and stacks, offering a pedagogical entry point for new- comers.

• Higher Topos Theory: Provides a detailed exposition of the transition from classical perspectives to the modern context of cohesive higher topos theory and global equivariant unstable homotopy theory.

• Synthetic Orbifold Geometry: Lays out a powerful synthetic differential theory of higher orbifold Cartan geometry, utilizing systems of modal operators to rigorously capture orbi-singularities and proper equivariant homotopy types.

• Unification of Methods: Unifies orbifold geometry with classical differential geometry and cohomology, compatibly enhancing the picture with modalities that reflect proper equivariant structures.

• Concrete Applications: Showcases the theory’s utility through the analysis of tangentially twisted nonabelian orbifold cohomology, specifically J-twisted orbifold Cohomotopy, applied to "Hypothesis H" and the classification of topo- logical phases.

This volume is designed to serve a dual purpose: the first part acts as an invitation for advanced graduate students and beyond, in mathematics and theoretical physics, leading them from basic topological charges to modern research applications. The subsequent parts provide a comprehensive resource for academic researchers in al- gebraic topology, differential geometry, and string theory interested in a cutting edge formulation of geometric cohomology and its applications to quantum systems.

Zielgruppe

Academic

Autoren/Hrsg.

Sati, Hisham

Schreiber, Urs

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Topologie

Inhaltsverzeichnis

1 Introduction. 2 Generalized Cohomology. 3 Phases & Branes. 4 Nonabelian Cohomology. 5 Orbifold K-Theory. 6 Geometric Homotopy. 7 Equivariant Homotopy. 8 Higher geometry. 9 Singular geometry.10 Orbifold Geometry.

Über Autor(innen)

Hisham Sati is Professor of Mathematics at NYU Abu Dhabi and the founding director of the Center for Quantum and Topological Systems. His interdisciplinary research spans mathematical physics, algebraic topology, and differential geometry, and their interactions through fundamental physical theories. He has delivered the Adams Memorial Lecture in Topology.

Urs Schreiber is Research Scientist at NYU Abu Dhabi, specializing in the mathe- matical foundations of quantum field theory. His work applies algebraic topology and geometric homotopy theory to fundamental physics, including topological quantum technology. He is the creator of the nLab, a research wiki for math and physics.