A Gentle Introduction to Homological Mirror Symmetry: 99 (London Mathematical Society Student Texts, Series Number 99)
Cambridge University Press, 8/19/2021
EAN 9781108483506, ISBN10: 110848350X
Hardcover, 400 pages, 23.5 x 15.8 x 2.9 cm
Language: English
Originally published in English
Homological mirror symmetry has its origins in theoretical physics but is now of great interest in mathematics due to the deep connections it reveals between different areas of geometry and algebra. This book offers a self-contained and accessible introduction to the subject via the representation theory of algebras and quivers. It is suitable for graduate students and others without a great deal of background in homological algebra and modern geometry. Each part offers a different perspective on homological mirror symmetry. Part I introduces the A-infinity formalism and offers a glimpse of mirror symmetry using representations of quivers. Part II discusses various A- and B-models in mirror symmetry and their connections through toric and tropical geometry. Part III deals with mirror symmetry for Riemann surfaces. The main mathematical ideas are illustrated by means of simple examples coming mainly from the theory of surfaces, helping the reader connect theory with intuition.
Part I. To A∞ and Beyond
1. Categories
2. Cohomology
3. Higher products
4. Quivers
Part II. A Glance through the Mirror
5. Motivation from physics
6. The A-side
7. The B-side
8. Mirror symmetry
Part III. Reflections on Surfaces
9. Gluing
10. Grading
11. Stabilizing
12. Deforming
References
Index.