Topology and Physics

Since its birth in Poincaré's seminal 1894 'Analysis Situs', topology has become a cornerstone of mathematics. As with all beautiful mathematical concepts, topology inevitably — resonating with that Wignerian principle of the effectiveness of mathematics in the natural sciences — finds its prominent role in physics. From Chern–Simons theory to topological quantum field theory, from knot invariants to Calabi–Yau compactification in string theory, from spacetime topology in cosmology to the recent Nobel Prize winning work on topological insulators, the interactions between topology and physics have been a triumph over the past few decades.

In this eponymous volume, we are honoured to have contributions from an assembly of grand masters of the field, guiding us with their world-renowned expertise on the subject of the interplay between 'Topology' and 'Physics'. Beginning with a preface by Chen Ning Yang on his recollections of the early days, we proceed to a novel view of nuclei from the perspective of complex geometry by Sir Michael Atiyah and Nick Manton, followed by an entrée toward recent developments in two-dimensional gravity and intersection theory on the moduli space of Riemann surfaces by Robbert Dijkgraaf and Edward Witten; a study of Majorana fermions and relations to the Braid group by Louis H Kauffman; a pioneering investigation on arithmetic gauge theory by Minhyong Kim; an anecdote-enriched review of singularity theorems in black-hole physics by Sir Roger Penrose; an adventure beyond anyons by Zhenghan Wang; an aperçu on topological insulators from first-principle calculations by Haijun Zhang and Shou-Cheng Zhang; finishing with synopsis on quantum information theory as one of the four revolutions in physics and the second quantum revolution by Xiao-Gang Wen. We hope that this book will serve to inspire the research community.

Contents:

• Preface — Early Examples of Topological Concepts in Physics (C N Yang)
• Complex Geometry of Nuclei and Atoms (M F Atiyah and N S Manton)
• Developments in Topological Gravity (Robbert Dijkgraaf and Edward Witten)
• Majorana Fermions and Representations of the Braid Group (Louis H Kauffman)
• Arithmetic Gauge Theory: A Brief Introduction (Minhyong Kim)
• Singularity Theorems (Roger Penrose)
• Beyond Anyons (Zhenghan Wang)
• Four Revolutions in Physics and the Second Quantum Revolution — A Unification of Force and Matter by Quantum Information (Xiao-Gang Wen)
• Topological Insulators from the Perspective of First-principles Calculations (Haijun Zhang and Shou-Cheng Zhang)
• Appendix — SO₄ symmetry in a Hubbard Model (C N Yang and Shou-Cheng Zhang)

Readership: Senior undergraduate student; graduate students; researchers.
Key Features:

• World-renowned mathematicians and physicists are the Chapter contributors
• It provides the latest review of the interplay between "Topology" and "Physics"
• Readers can appreciate the beauty of mathematics and physics within one single book