A gentle introduction to homological mirror symmetry / Raf Bocklandt, University of Amsterdam.

Format:

Book

Author/Creator:

Bocklandt, Raf, 1977- author.

Series:

London Mathematical Society student texts ; 99.

London Mathematical Society student texts ; 99

Language:

English

Subjects (All):

Mirror symmetry.

Homology theory.

Physical Description:

1 online resource (xi, 390 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2021.

Summary:

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.

Contents:

Cover

Series information

Title page

Copyright information

Contents

Preface

PART ONE TO A[sub(infty)] AND BEYOND

1 Categories

1.1 Categories

1.2 Functors

1.3 Natural Transformations

1.4 Linear Categories

1.5 Modules

1.6 Morita Equivalence

1.7 Exercises

2 Cohomology

2.1 Complexes

2.2 Cohomology in Topology

2.3 Cohomology in Algebra

2.4 Exercises

3 Higher Products

3.1 Motivation and Definition

3.2 Minimal Models

3.3 A[sub(infty)]-Categories

3.4 Bells and Whistles

3.5 Exercises

4 Quivers

4.1 Representations of Quivers

4.2 Strings and Bands

4.3 Points and Sheaves

4.4 Picturing the Categories

4.5 A First Glimpse of Homological Mirror Symmetry

4.6 Exercises

PART TWO A GLANCE THROUGH THE MIRROR

5 Motivation from Physics

5.1 The Path Integral Formalism

5.2 Symmetry

5.3 Superstrings

5.4 Categorical Interpretations

5.5 What Is Mirror Symmetry?

5.6 Exercises

6 The A-Side

6.1 Morse Theory

6.2 The Basic Fukaya Category

6.3 Variations

6.4 Generators

6.5 Exercises

7 The B-Side

7.1 Varieties

7.2 Other Geometrical Objects

7.3 Equivalences

7.4 Exercises

8 Mirror Symmetry

8.1 The Complex Torus

8.2 Toric Varieties

8.3 Tropical Geometry

8.4 One, Two, Three, Mirror Symmetry

8.5 Away from the Large Limit

8.6 Mirrors and Fibrations

8.7 Exercises

PART THREE REFLECTIONS ON SURFACES

9 Gluing

9.1 Marked Surfaces

9.2 Gluing Arcs to Strings and Bands

9.3 Gluing Fukaya Categories over Graphs

9.4 Gluing and Mirror Symmetry

9.5 Covers

9.6 Dimer Models

9.7 Mirrors Galore

9.8 Exercises

10 Grading

10.1 Graded Surfaces

10.2 Strings and Bands

10.3 Characterizing Graded Surfaces

10.4 Gradings and Matrix Factorizations

10.5 Mirror Varieties

10.6 Mirror Orbifolds.

10.7 Exercises

11 Stabilizing

11.1 The Grothendieck Group

11.2 King Stability and Pair of Pants Decompositions

11.3 Bridgeland Stability

11.4 Stability Conditions and Quadratic Differentials

11.5 Stability Manifolds

11.6 Exercises

12 Deforming

12.1 Deformation Theory for A[sub(infty)]-Algebras

12.2 A[sub(infty)]-Extensions

12.3 Deformation Theory for Gentle Algebras

12.4 Filling the Pair of Pants

12.5 Koszul Duality

12.6 The Mirror Functor

12.7 Exercises

References

Index.

Notes:

Title from publisher's bibliographic system (viewed on 20 Aug 2021).

ISBN:

1-108-69245-1

OCLC:

1261767703