Instanton Counting, Quantum Geometry and Algebra

Description

This book pedagogically describes recent developments in gauge theory, in particular four-dimensional N = 2 supersymmetric gauge theory, in relation to various fields in mathematics, including algebraic geometry, geometric representation theory, vertex operator algebras. The key concept is the instanton, which is a solution to the anti-self-dual Yang–Mills equation in four dimensions. In the first part of the book, starting with the systematic description of the instanton, how to integrate out the instanton moduli space is explained together with the equivariant localization formula. It is then illustrated that this formalism is generalized to various situations, including quiver and fractional quiver gauge theory, supergroup gauge theory. The second part of the book is devoted to the algebraic geometric description of supersymmetric gauge theory, known as the Seiberg–Witten theory, together with string/M-theory point of view. Based on its relation to integrable systems, how to quantize such a geometric structure via the Ω-deformation of gauge theory is addressed. The third part of the book focuses on the quantum algebraic structure of supersymmetric gauge theory. After introducing the free field realization of gauge theory, the underlying infinite dimensional algebraic structure is discussed with emphasis on the connection with representation theory of quiver, which leads to the notion of quiver W-algebra. It is then clarified that such a gauge theory construction of the algebra naturally gives rise to further affinization and elliptic deformation of W-algebra. 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 θ-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 4.2 Seiberg–Witten Theory 4.2.1 Renormalization Group Analysis 4.2.2 One-Loop Exactness 4.2.3 SU(2) Theory 4.2.4 SU(n) Theory 4.2.5 mathcalN = 2 SQCD 4.3 Quiver Gauge Theory 4.3.1 A1 Quiver 4.3.2 A2 Quiver 4.3.3 A3 Quiver 4.3.4 Generic Quiver 4.4 Supergroup Gauge Theory 4.5 Brane Dynamics and mathcalN = 2 Gauge Theory 4.5.1 Hanany–Witten Construction 4.5.2 Seiberg–Witten Curve from M-Theory 4.5.3 Quiver Gauge Theory 4.5.4 Higgsing and Vortices 4.5.5 Higgsing in Seiberg–Witten Geometry 4.5.6 Supergroup Gauge Theory 4.6 Eight Supercharge Theory in Higher Dimensions 4.6.1 5d mathcalN = 1 Theory 4.6.2 6d mathcalN = (1,0) Theory References 5 Quantization of Geometry 5.1 Non-perturbative Schwinger–Dyson Equation 5.1.1 Add/remove Instantons 5.2 qq-Character 5.2.1 iWeyl Reflection 5.2.2 Supergroup Gauge Theory 5.2.3 Higher Weight Current 5.2.4 Collision Limit 5.3 Classical Limit 5.3.1 (Very) Classical Limit: ε1,2 to0 5.3.2 Nekrasov–Shatashvili Limit: ε2 to0 5.4 Examples 5.4.1 A1 Quiver 5.4.2 A2 Quiver 5.4.3 Affine A0 quiver 5.5 Gauge Origami Reloaded 5.5.1 8d Gauge Origami Partition Function 5.5.2 qq-Character Integral Formula 5.6 Quantization of Cycle Integrals 5.6.1 Saddle Point Equation 5.6.2 Y-Function 5.7 Quantum Geometry and Quantum Integrability 5.7.1 Pure SU(n) Yang–Mills Theory 5.7.2 mathcalN = 2 SQCD 5.7.3 A2 Quiver 5.7.4 Ap Quiver 5.8 Bethe Equation 5.8.1 Saddle Point Equation 5.8.2 Higgsing and Truncation 5.8.3 Dimensional Hierarchy: Periodicity of Spectral Parameter References Part III Quantum Algebra 6 Operator Formalism of Gauge Theory 6.1 Holomorphic Deformation 6.1.1 Free Field Realization 6.2 Z-state 6.2.1 Screening Current 6.2.2 Instanton Sum and Screening Charge 6.2.3 V-operator: Fundamental Matter 6.2.4 Boundary Degrees of Freedom 6.2.5 Y-Operator: Observable Generator 6.2.6 A-operator: iWeyl Reflection Generator 6.3 Pole Cancellation Mechanism References 7 Quiver W-Algebra 7.1 T-Operator: Generating Current 7.2 Classical Limit: Quantum Integrability 7.3 Examples 7.3.1 A1 Quiver 7.3.2 A2 Quiver 7.3.3 Ap Quiver 7.3.4 Dp Quiver 7.4 Fractional Quiver W-Algebra 7.4.1 Screening Current 7.4.2 Y-Operator 7.4.3 A-Operator 7.4.4 iWeyl Reflection 7.4.5 T-operator: Generating Current 7.4.6 BC2 Quiver 7.4.7 Bp Quiver 7.4.8 Cp Quiver 7.4.9 G2 Quiver 7.4.10 NS1,2 Limit 7.5 Affine Quiver W-Algebra 7.5.1 Affine A0 quiver 7.5.2 Affine Ap-1 quiver 7.6 Integrating over Quiver Variety 7.6.1 Instanton Partition Function 7.6.2 qq-Character References 8 Quiver Elliptic W-Algebra 8.1 Operator Formalism 8.1.1 Doubled Fock Space 8.1.2 Screening Current 8.1.3 Z-State 8.2 Trace Formula 8.2.1 Coherent State Basis 8.2.2 Torus Correlation Function 8.2.3 Connection to Elliptic Quantum Group 8.3 More on Elliptic Vertex Operators 8.3.1 V-Operator 8.3.2 Y-Operator 8.3.3 A-Operator 8.4 T-Operator 8.4.1 A1 Quiver 8.4.2 A2 Quiver References Appendix A Special Functions A.1 Gamma Functions A.1.1 Reflection Formula A.1.2 Multiple Sine Function A.2 q-Functions A.2.1 q-Shifted Factorial A.2.2 Quantum Dilogarithm A.2.3 q-Gamma Functions A.2.4 Partition Sum A.3 Elliptic Functions A.3.1 Theta Function A.3.2 Elliptic Gamma Functions A.3.3 Elliptic Analog of Polylogarithm Appendix B Combinatorial Calculus B.1 Partition B.2 Instanton Calculus B.2.1 U(n) Theory B.2.2 U(n0|n1) Theory Appendix C Matrix Model C.1 Matrix Integral C.1.1 Eigenvalue Integral Representation C.2 Saddle Point Analysis C.2.1 Eigenvalue Density Function C.2.2 Functional Representation C.3 Spectral Curve C.3.1 Cycle Integrals C.4 Quantum Geometry C.4.1 Baker–Akhiezer Function C.4.2 Quantization of the Cycle C.5 Quantum Algebra C.5.1 Loop Equation C.5.2 Operator Formalism C.5.3 Gauge Theory Parameter C.5.4 Vertex Operators C.5.5 Z-State Index