Contents

0 Introduction
1 Electromagnetic duality and monopoles
 1.1 Electric and magnetic charges
 1.2 The S and the T transformations
 1.3 ’t Hooft-Polyakov monopoles
  1.3.1 Classical features
  1.3.2 Semiclassical features
2 𝒩=2 multiplets and Lagrangians
 2.1 Microscopic Lagrangian
  2.1.1 𝒩=1 superfields
  2.1.2 Vector multiplets and hypermultiplets
 2.2 Vacua
 2.3 BPS bound
 2.4 Low energy Lagrangian
3 Renormalization and anomaly
 3.1 Renormalization
 3.2 Anomalies
  3.2.1 Anomalies of global symmetry
  3.2.2 Anomalies of gauge symmetry
 3.3 𝒩=1 pure Yang-Mills
  3.3.1 Confinement and gaugino condensate
  3.3.2 The theory in a box
4 Seiberg-Witten solution to pure SU(2) theory
 4.1 One-loop running and the monodromy at infinity
 4.2 Behavior in the strongly-coupled region
 4.3 The Seiberg-Witten solution
  4.3.1 The curve
  4.3.2 The monodromy M
  4.3.3 The monodromies M±
 4.4 Less supersymmetric cases
  4.4.1 𝒩=1 system
  4.4.2 Pure bosonic system
 4.5 SU(2) vs SO(3)
5 SU(2) theory with one flavor
 5.1 Structure of the u-plane
  5.1.1 Schematic running of the coupling
  5.1.2 Monodromies
 5.2 The curve
 5.3 Some notable features
6 Curves and 6d 𝒩=(2, 0) theory
 6.1 Strings with variable tension
 6.2 Strings with variable tension from membranes
  6.2.1 General idea
  6.2.2 Example: pure SU(2) theory
 6.3 Self-duality of the 6d theory
 6.4 Intermediate 5d Yang-Mills theory and its boundary conditions
  6.4.1 Five-dimensional maximally-supersymmetric Yang-Mills
  6.4.2 𝒩=4 super Yang-Mills
  6.4.3 𝒩=2 pure SU(2) theory and the Nf = 1 theory
  6.4.4 The SU(2) theories with Nf = 2, 3, 4
7 Higgs branches and hyperkähler manifolds
 7.1 General structures of the Higgs branch Lagrangian
 7.2 Hypermultiplets revisited
 7.3 The hyperkähler quotient
  7.3.1 U(1) gauge theory with one charged hypermultiplet
  7.3.2 SU(2) gauge theory with two hypermultiplets in the doublet
8 SU(2) theory with 2 and 3 flavors
 8.1 Generalities
 8.2 Nf = 2: the curve and the monodromies
 8.3 Nf = 2: the discrete R-symmetry
 8.4 Nf = 2: the moduli space
 8.5 Nf = 3
9 SU(2) theory with 4 flavors and Gaiotto’s duality
 9.1 The curve as λ2 = ϕ2(z)
 9.2 Identification of parameters
  9.2.1 Coupling constant
  9.2.2 Mass parameters
 9.3 Weak-coupling limit and trifundamentals
 9.4 Strong-coupling limit
 9.5 Generalization
  9.5.1 Trivalent diagrams
  9.5.2 Example: torus with one puncture
  9.5.3 Example: sphere with five punctures
  9.5.4 Example: a genus-two surface
  9.5.5 The curve and the Hitchin field
 9.6 Theories with less flavors revisited
  9.6.1 Rewriting of the curves
  9.6.2 Generalization
10 Argyres-Douglas CFTs
 10.1 Nf = 1 theory and the simplest Argyres-Douglas CFT
 10.2 Argyres-Douglas CFT from the Nf = 2 theory
 10.3 Argyres-Douglas CFT from the Nf = 3 theory
 10.4 Summary of rank-1 theories
  10.4.1 Argyres-Douglas CFTs from SU(2) with flavors
  10.4.2 Exceptional theories of Minahan-Nemeschansky
  10.4.3 Newer rank-1 theories
 10.5 More general Argyres-Douglas CFTs: XN and Y N
11 Theories with other simple gauge groups
 11.1 Semiclassical analysis
 11.2 Pure SU(N) theory
  11.2.1 The curve
  11.2.2 Infrared gauge coupling matrix
 11.3 SU(N) theory with fundamental flavors
  11.3.1 Nf = 1
  11.3.2 General number of flavors
 11.4 SO(2N) theories
  11.4.1 Semi-classical analysis
  11.4.2 Pure SO(2N) theory
  11.4.3 SO(2N) theory with flavors in the vector representation
 11.5 Argyres-Douglas CFTs
  11.5.1 Pure SU(N) theory
  11.5.2 SU(N) theory with two flavors
  11.5.3 Pure SO(2N) theory
  11.5.4 Argyres-Douglas CFTs and the Higgs branch
 11.6 Seiberg-Witten solutions for various other simple gauge groups
12 Argyres-Seiberg-Gaiotto duality for SU(N) theory
 12.1 S-dual of SU(N) with Nf = 2N flavors, part I
  12.1.1 Rewriting of the curve
  12.1.2 Weak-coupling limit
  12.1.3 A strong-coupling limit
 12.2 SU(N) quiver theories and tame punctures
  12.2.1 Quiver gauge theories
  12.2.2 𝒩=2 theory
  12.2.3 Linear quiver theories
  12.2.4 Tame punctures
  12.2.5 Tame punctures and the number of Coulomb branch operators
  12.2.6 Tame punctures and the decoupling
 12.3 S-dual of SU(N) with Nf = 2N flavors, part II
  12.3.1 For general N
  12.3.2 N = 3: Argyres-Seiberg duality
 12.4 Applications
  12.4.1 TN
  12.4.2 MN(E7)
  12.4.3 MN(E8)
  12.4.4 The singular limit of SU(N) with even number of flavors
 12.5 Tame punctures and Higgsing
13 Conclusions and further directions