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Table of Contents
Intro
Preface
Gauge Theory in Physics and Mathematics
Universality of QFT
mathcalN=2 Supersymmetry
Instanton Counting
Seiberg-Witten Theory
Relation to Integrable System
Quantization of Geometry
Quantum Algebraic Structure
Quiver W-algebra
References
Acknowledgements
Contents
Part I Instanton Counting
1 Instanton Counting and Localization
1.1 Yang-Mills Theory
1.2 Instanton
1.3 Summing up Instantons
1.3.1)-Term
1.3.2 Topological Twist
1.4 ADHM Construction of Instantons
1.4.1 ADHM Equation
1.4.2 Constructing Instanton
1.4.3 Dirac Zero Mode
1.4.4 String Theory Perspective
1.5 Instanton Moduli Space
1.5.1 Compactification and Resolution
1.5.2 Stability Condition
1.6 Equivariant Localization of Instanton Moduli Space
1.6.1 Equivariant Cohomology
1.6.2 Equivariant Localization
1.6.3 Equivariant Action and Fixed Point Analysis
1.7 Integrating ADHM Variables
1.7.1 Path Integral Formalism
1.7.2 Contour Integral Formula
1.7.3 Incorporating Matter
1.7.4 Pole Analysis
1.8 Equivariant Index Formula
1.8.1 Spacetime Bundle
1.8.2 Framing and Instanton Bundles
1.8.3 Universal Bundle
1.8.4 Index Formula
1.8.5 Vector Multiplet
1.8.6 Fundamental and Antifundamental Matters
1.8.7 Adjoint Matter
1.9 Instanton Partition Function
1.9.1 Vector Multiplet
1.9.2 Fundamental and Antifundamental Matters
1.9.3 Adjoint Matter
1.9.4 Chern-Simons Term
1.9.5 Relation to the Contour Integral Formula
References
2 Quiver Gauge Theory
2.1 Instanton Moduli Space
2.1.1 Vector Bundles on the Moduli Space
2.1.2 Equivariant Fixed Point and Observables
2.2 Instanton Partition Function
2.2.1 Equivariant Index Formula
2.2.2 Contour Integral Formula
2.2.3 Quiver Cartan Matrix
2.3 Quiver Variety
2.3.1 ADHM Quiver
2.3.2 ADHM on ALE Space
2.3.3 Gauge Origami
2.4 Fractional Quiver Gauge Theory
2.4.1 Instanton Moduli Space
2.4.2 Instanton Partition Function
References
3 Supergroup Gauge Theory
3.1 Supergroup Yang-Mills Theory
3.1.1 Supervector Space, Superalgebra, and Supergroup
3.1.2 Yang-Mills Theory
3.1.3 Quiver Gauge Theory Description
3.2 Decoupling Trick
3.2.1 Vector Multiplet
3.2.2 Bifundamental Hypermultiplet
3.2.3 Dp Quiver
3.2.4 Affine A0 quiver
3.3 ADHM Construction of Super Instanton
3.3.1 ADHM Data
3.3.2 Constructing Instanton
3.3.3 String Theory Perspective
3.3.4 Instanton Moduli Space
3.4 Equivariant Localization
3.4.1 Framing and Instanton Bundles
3.4.2 Observable Bundles
3.4.3 Equivariant Index Formula
3.4.4 Instanton Partition Function
3.4.5 Contour Integral Formula
References
Part II Quantum Geometry
4 Seiberg-Witten Geometry
4.1 mathcalN = 2 Gauge Theory in Four Dimensions
4.1.1 Supersymmetric Vacua
4.1.2 Low Energy Effective Theory
4.1.3 BPS Spectrum
Preface
Gauge Theory in Physics and Mathematics
Universality of QFT
mathcalN=2 Supersymmetry
Instanton Counting
Seiberg-Witten Theory
Relation to Integrable System
Quantization of Geometry
Quantum Algebraic Structure
Quiver W-algebra
References
Acknowledgements
Contents
Part I Instanton Counting
1 Instanton Counting and Localization
1.1 Yang-Mills Theory
1.2 Instanton
1.3 Summing up Instantons
1.3.1)-Term
1.3.2 Topological Twist
1.4 ADHM Construction of Instantons
1.4.1 ADHM Equation
1.4.2 Constructing Instanton
1.4.3 Dirac Zero Mode
1.4.4 String Theory Perspective
1.5 Instanton Moduli Space
1.5.1 Compactification and Resolution
1.5.2 Stability Condition
1.6 Equivariant Localization of Instanton Moduli Space
1.6.1 Equivariant Cohomology
1.6.2 Equivariant Localization
1.6.3 Equivariant Action and Fixed Point Analysis
1.7 Integrating ADHM Variables
1.7.1 Path Integral Formalism
1.7.2 Contour Integral Formula
1.7.3 Incorporating Matter
1.7.4 Pole Analysis
1.8 Equivariant Index Formula
1.8.1 Spacetime Bundle
1.8.2 Framing and Instanton Bundles
1.8.3 Universal Bundle
1.8.4 Index Formula
1.8.5 Vector Multiplet
1.8.6 Fundamental and Antifundamental Matters
1.8.7 Adjoint Matter
1.9 Instanton Partition Function
1.9.1 Vector Multiplet
1.9.2 Fundamental and Antifundamental Matters
1.9.3 Adjoint Matter
1.9.4 Chern-Simons Term
1.9.5 Relation to the Contour Integral Formula
References
2 Quiver Gauge Theory
2.1 Instanton Moduli Space
2.1.1 Vector Bundles on the Moduli Space
2.1.2 Equivariant Fixed Point and Observables
2.2 Instanton Partition Function
2.2.1 Equivariant Index Formula
2.2.2 Contour Integral Formula
2.2.3 Quiver Cartan Matrix
2.3 Quiver Variety
2.3.1 ADHM Quiver
2.3.2 ADHM on ALE Space
2.3.3 Gauge Origami
2.4 Fractional Quiver Gauge Theory
2.4.1 Instanton Moduli Space
2.4.2 Instanton Partition Function
References
3 Supergroup Gauge Theory
3.1 Supergroup Yang-Mills Theory
3.1.1 Supervector Space, Superalgebra, and Supergroup
3.1.2 Yang-Mills Theory
3.1.3 Quiver Gauge Theory Description
3.2 Decoupling Trick
3.2.1 Vector Multiplet
3.2.2 Bifundamental Hypermultiplet
3.2.3 Dp Quiver
3.2.4 Affine A0 quiver
3.3 ADHM Construction of Super Instanton
3.3.1 ADHM Data
3.3.2 Constructing Instanton
3.3.3 String Theory Perspective
3.3.4 Instanton Moduli Space
3.4 Equivariant Localization
3.4.1 Framing and Instanton Bundles
3.4.2 Observable Bundles
3.4.3 Equivariant Index Formula
3.4.4 Instanton Partition Function
3.4.5 Contour Integral Formula
References
Part II Quantum Geometry
4 Seiberg-Witten Geometry
4.1 mathcalN = 2 Gauge Theory in Four Dimensions
4.1.1 Supersymmetric Vacua
4.1.2 Low Energy Effective Theory
4.1.3 BPS Spectrum