Quantum Geometry, Matrix Theory, and Gravity

Building on mathematical structures familiar from quantum mechanics, this book provides an introduction to quantization in a broad context before developing a framework for quantum geometry in Matrix Theory and string theory. Taking a physics-oriented approach to quantum geometry, this framework helps explain the physics of Yang–Mills-type matrix models, leading to a quantum theory of space-time and matter. This novel framework is then applied to Matrix Theory, which is defined through distinguished maximally supersymmetric matrix models related to string theory. A mechanism for gravity is discussed in depth, which emerges as a quantum effect on quantum space-time within Matrix Theory. Using explicit examples and exercises, readers will develop a physical intuition for the mathematical concepts and mechanisms. It will benefit advanced students and researchers in theoretical and mathematical physics, and is a useful resource for physicists and mathematicians interested in the geometrical aspects of quantization in a broader context.

Table of Contents:
Preface; The trouble with spacetime; Quantum geometry and Matrix theory; Part I. Mathematical Background: 1. Differentiable manifolds; 2. Lie groups and coadjoint orbits; Part II. Quantum Spaces and Geometry: 3. Quantization of symplectic manifolds; 4. Quantum spaces and matrix geometry; 5. Covariant quantum spaces; Part III. Noncommutative field theory and matrix models: 6. Noncommutative field theory; 7. Yang–Mills matrix models and quantum spaces; 8. Fuzzy extra dimensions; 9. Geometry and dynamics in Yang–Mills matrix models; 10. Higher-spin gauge theory on quantum spacetime; Part IV. Matrix Theory and Gravity: 11. Matrix theory: maximally supersymmetric matrix models; 12. Gravity as a quantum effect on quantum spacetime; 13. Matrix quantum mechanics and the BFSS model; Appendixes; References; Index.

About the Author :
Harold Steinacker is senior scientist at the University of Vienna. He obtained his Ph.D. in physics at the University of California at Berkeley, and has held research positions at several universities. He has published more than 100 research papers, contributing significantly to the understanding of quantum geometry and matrix models in fundamental physics.

Review :
'This text provides an invaluable introduction to quantum spaces, quantum geometry and matrix models, culminating in in-depth discussions of the IKKT and BFSS matrix models, proposed non-perturbative definitions of superstring theory. I highly recommend this book to anyone seriously interested in these topics.' Robert Brandenberger, McGill University 'Based on the author's renowned expertise, this insightful masterpiece delves into noncommutative geometry, matrix models, and their role in string theory and quantum gravity. The book is consistently written from a physics viewpoint with specific examples, offering fresh perspectives and suggesting fascinating possibilities for novice and seasoned researchers alike.' Hikaru Kawai, National Taiwan University 'The first complete book that puts together over 25 years of contemporary research connecting noncommutative field theory with gravity. Starting at a suitably pedagogical level for use as a textbook in an advanced graduate-level physics course, it elucidates state-of-the-art developments, making it an invaluable reference source for both novices and experts.' Richard J. Szabo, Heriot-Watt University 'Current approaches to the great puzzle of quantum gravity, with demonstrated potential for success, include string theory (or M-theory) and noncommutative geometry. This book masterfully brings together these two approaches, providing a perspective as well as background material. It will be a valuable asset to researchers in quantum gravity.' Parameswaran Nair, The City College of New York 'This text provides perhaps the first bridge for the uninitiated into this interesting field of research. The introductory chapters of this text are perhaps its finest, which beautifully define and explain the essential concepts needed for the proceeding discussions. Moreover, this book provides more than just a broad survey of historical results in the field, in the latter quadrants of the text, recent research is outlined in full, giving the reader a current outline of the field.' Physics Book Reviewer