The maximally supersymmetric theory

For more info about our services contact : help@bestpfe.com

Table of contents

1 Presentation 
1.1 Yang-Mills Gauge Theories in Four Dimensions
1.1.1 Classical Electric-Magnetic Duality
1.1.2 Non-perturbative Physics
1.1.3 Phases of Yang-Mills Theories
1.1.4 Line Operators
1.1.5 The Global Structure of the Gauge Group
1.2 Supersymmetric Gauge Theories and Modularity
1.2.1 The maximally supersymmetric theory
1.2.2 From S-duality to modular invariance
1.2.3 The vacuum structure
1.2.4 Argyres-Douglas points
1.2.5 Compactification on a cylinder
1.3 Vacua of the N = 1∗ Theory: a Summary
2 Massive Vacua of N = 1∗ Theory on R4 
2.1 Introduction
2.2 The N = 1∗ Theory on R4
2.2.1 The N = 4 Gauge Theory
2.2.2 The Mass Deformation
2.2.3 Field Theoretic Properties of N = 1∗
2.3 The Vacuum Structure
2.3.1 Classical Vacua
2.3.2 The Witten Index
2.3.3 Quantum Vacua
2.3.4 Modularity of the Theory
2.4 The Semi-Classical Configurations and the Classification Problem
2.4.1 Semi-classical Configurations and sl(2) algebras
2.4.2 The Gauge Group, Triples and Nilpotent Orbits
2.4.3 Nilpotent Orbit Theory
2.4.4 Number of Partitions
2.4.5 The Centralizer and the Index
2.5 The Counting for the Classical Groups
2.5.1 The Nilpotent Orbits
2.5.2 The Centralizers
2.5.3 The Supersymmetric Index for the Classical Groups
2.5.4 The Generating Functions
2.5.5 Illustrative Examples
2.6 The Counting for the Exceptional Groups
2.6.1 The Orbits and the Centralizers
2.6.2 The Supersymmetric Index for the Exceptional Groups
3 Elliptic Integrable Systems and Modularity 
3.1 Introduction
3.2 The Calogero-Moser system
3.2.1 General Definitions
3.2.2 Why Calogero-Moser Systems ?
3.2.3 Classical and Quantum Integrability
3.2.4 Twisted and Untwisted Elliptic Calogero-Moser Models
3.2.5 Complexified Models
3.2.6 The Symmetries of the Potential
3.3 Elliptic Integrable Systems and Modularity
3.3.1 Langlands Duality
3.3.2 Langlands Duality at Rank Two
3.4 Semi-Classical Limits of Elliptic Integrable Systems
3.4.1 Calogero-Moser systems and Toda systems
3.4.2 The Dual Affine Algebra and Non-Perturbative Contributions
3.4.3 Semi-Classical Limits
3.4.4 The Trigonometric, Affine Toda and Intermediate Limits
4 Isolated Extrema of the Twisted Elliptic Calogero-Moser System 
4.1 The Case AN−1 = su(N)
4.2 The B,C,D Models
4.3 The Case C2 = sp(4) = so(5) and Vector Valued Modular Forms
4.3.1 The Positions of the Extrema
4.3.2 Series Expansions of the Extrema
4.3.3 Modular Forms of the Hecke Group and the Γ0(4) Subgroup
4.3.4 A Remark on a Manifold of Extrema
4.4 The Case D4 = so(8) and the Point of Monodromy
4.4.1 The Singlet
4.4.2 The Triplet
4.4.3 The Quadruplet
4.4.4 The Duodecuplet and a Point of Monodromy
4.4.5 The List of Extrema for so(8)
4.5 The Dual Cases B3 = so(7) and C3 = sp(6)
4.5.1 Exact Multiplets
4.5.2 The Duodecuplet, the Quattuordecuplet and the Points of Monodromy .
4.5.3 The List of Extrema for so(7) and sp(6)
4.6 Partial Results for Other Lie Algebras
4.7 Conclusions
5 The N = 1∗ Gauge Theories on R3 × S1 
5.1 Supersymmetric gauge theories and integrable systems
5.1.1 Seiberg-Witten curves for N = 2 theories
5.1.2 Integrable Systems and N = 2 theories
5.1.3 The Calogero-Moser poteneial and N = 2∗
5.1.4 Breaking N = 2 to N = 1
5.2 The non-perturbative superpotential
5.2.1 A Strategy : Compactification
5.2.2 The gauge field on the cylinder
5.2.3 The Non-perturbative Superpotential for N = 1 theories
5.2.4 Electric-Magnetic Duality
5.3 The Massive Vacua of N = 1∗ gauge theories
5.3.1 Comparison with Vacua on R4
5.3.2 Tensionless Domain Walls, Colliding Quantum Vacua and Masslessness .
6 The SU(N) theory 
6.1 Introduction
6.2 Semi-Classical Preliminaries
6.3 The Gauge Algebra su(3), the Massless Branch and the Singularity
6.3.1 Semi-classical analysis
6.3.2 The Massless Branch and the Singularity
6.3.3 The Massless Branch in the Toroidal Variables
6.3.4 The points λ6 6= −1
6.3.5 The Z2 symmetric points
6.3.6 Dualities at the Z2 symmetric points
6.3.7 Summary Remarks
6.4 The Gauge Algebra su(4)
6.4.1 The Partition 2 + 1 + 1
6.4.2 The Partition 3 + 1
6.4.3 The Duality Diagram
6.4.4 Summary Remarks
6.5 A Word on the su(N) Theory
7 Topological properties of Groups and Lines 
7.1 Introduction
7.2 The N = 1∗ Theory with Gauge Algebra so(5)
7.2.1 The Semi-Classical Analysis and Nilpotent Orbit Theory
7.2.2 The Elliptic Integrable System
7.2.3 Global Properties of the Gauge Group and Line Operators
7.2.4 Summary and Motivation
7.2.5 Representations of the Vacua for B2 Theories
7.3 More on Nilpotent Orbit Theory
7.3.1 The Nilpotent Orbit Theory of Bala-Carter and Sommers
7.3.2 The Bridge between Gauge Theory and Integrable System
7.4 Discrete Gauge Groups and Wilson Lines
7.4.1 Discrete Gauge Groups and Wilson Lines
7.4.2 The Semi-Classical Vacua for G2
7.4.3 The Elliptic Integrable System and the Semi-classical Limits
7.4.4 Results Based on Numerics
7.4.5 Langlands Duality and the Duality Diagram
7.5 The so(5) Massless Branch
7.5.1 The Local Description of the Massless Branch
7.5.2 Duality and the Massless Branch
7.5.3 The Moduli Space of Vacua for the Different Gauge Theories
A Supersymmetric gauge theories 
A.1 Supersymmetry in various dimensions
A.2 Supersymmetric gauge theories in four dimensions
A.2.1 The non supersymmetric gauge theories
A.2.2 The N = 1 gauge theory
A.2.3 The N = 2 gauge theory
A.2.4 The N = 4 gauge theory
B Lie Algebra 
B.1 Basic definitions
B.2 Lattices
B.3 Lie algebra data
C Modular Forms and Elliptic Functions 
C.1 Modular and Automorphic forms
C.1.1 Definitions
C.1.2 Eisenstein Series
C.2 Elliptic functions
C.2.1 The Weierstrass function
C.2.2 The twisted Weierstrass functions
C.3 Theta and Eta Functions
C.4 Modular Forms and Sublattices