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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

$S U (2)$ 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

$M_{\infty}$
4.3.3 The monodromies

$M_{\pm}$
4.4 Less supersymmetric cases
4.4.1

$𝒩 = 1$ system
4.4.2 Pure bosonic system
4.5

$S U (2)$ vs

$S O (3)$
5

$S U (2)$ 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

$S U (2)$ 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

$S U (2)$ theory and the

$N_{f} = 1$ theory
6.4.4 The

$S U (2)$ theories with

$N_{f} = 2, 3, 4$
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

$S U (2)$ gauge theory with two hypermultiplets in the doublet
8

$S U (2)$ theory with 2 and 3 ﬂavors
8.1 Generalities
8.2

$N_{f} = 2$ : the curve and the monodromies
8.3

$N_{f} = 2$ : the discrete R-symmetry
8.4

$N_{f} = 2$ : the moduli space
8.5

$N_{f} = 3$
9

$S U (2)$ theory with 4 ﬂavors and Gaiotto’s duality
9.1 The curve as

$λ^{2} = ϕ_{2} (z)$
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

$N_{f} = 1$ theory and the simplest Argyres-Douglas CFT
10.2 Argyres-Douglas CFT from the

$N_{f} = 2$ theory
10.3 Argyres-Douglas CFT from the

$N_{f} = 3$ theory
10.4 Summary of rank-1 theories
10.4.1 Argyres-Douglas CFTs from

$S U (2)$ with ﬂavors
10.4.2 Exceptional theories of Minahan-Nemeschansky
10.4.3 Newer rank-1 theories
10.5 More general Argyres-Douglas CFTs:

$X_{N}$ and

$Y_{N}$
11 Theories with other simple gauge groups
11.1 Semiclassical analysis
11.2 Pure

$S U (N)$ theory
11.2.1 The curve
11.2.2 Infrared gauge coupling matrix
11.3

$S U (N)$ theory with fundamental ﬂavors
11.3.1

$N_{f} = 1$
11.3.2 General number of ﬂavors
11.4

$S O (2 N)$ theories
11.4.1 Semi-classical analysis
11.4.2 Pure

$S O (2 N)$ theory
11.4.3

$S O (2 N)$ theory with ﬂavors in the vector representation
11.5 Argyres-Douglas CFTs
11.5.1 Pure

$S U (N)$ theory
11.5.2

$S U (N)$ theory with two ﬂavors
11.5.3 Pure

$S O (2 N)$ 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

$S U (N)$ theory
12.1 S-dual of

$S U (N)$ with

$N_{f} = 2 N$ 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

$S U (N)$ 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

$S U (N)$ with

$N_{f} = 2 N$ ﬂavors, part II
12.3.1 For general

$N$
12.3.2

$N = 3$ : Argyres-Seiberg duality
12.4 Applications
12.4.1

$T_{N}$
12.4.2

$M N (E_{7})$
12.4.3

$M N (E_{8})$
12.4.4 The singular limit of

$S U (N)$ with even number of ﬂavors
12.5 Tame punctures and Higgsing
13 Conclusions and further directions

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