IPMU 13-0234
UT-13-42
version 1

$𝒩 = 2$ supersymmetric dynamics for dummies

Yuji Tachikawa $^{♯, ♭}$

$^{♭}$	Department of Physics, Faculty of Science,
	University of Tokyo, Bunkyo-ku, Tokyo 133-0022, Japan
$^{♯}$	Kavli Institute for the Physics and Mathematics of the Universe (WPI),
	University of Tokyo, Kashiwa, Chiba 277-8583, Japan

abstract

We give a pedagogical introduction to the dynamics of $𝒩 = 2$ supersymmetric systems in four dimensions. The topic ranges from the Lagrangian and the Seiberg-Witten solutions of $S U (2)$ gauge theories to Argyres-Douglas CFTs and Gaiotto dualities, while discussions on instanton integrals are intentionally left out.

This lecture note is a write-up of the author’s lectures at Tohoku University, Nagoya University and Rikkyo University.

Contents
0 Introduction
Prerequisites, disclaimer, and acknowledgments
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
2

𝒩 = 2

multiplets and Lagrangians
2.1 Microscopic Lagrangian
2.2 Vacua
2.3 BPS bound
2.4 Low energy Lagrangian
3 Renormalization and anomaly
3.1 Renormalization
3.2 Anomalies
3.3

𝒩 = 1

pure Yang-Mills
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
4.4 Less supersymmetric cases
4.5

S U (2)

S O (3)

5

S U (2)

theory with one ﬂavor
5.1 Structure of the

u

-plane
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
6.3 Self-duality of the 6d theory
6.4 Intermediate 5d Yang-Mills theory and its boundary conditions
7 Higgs branches and hyperkähler manifolds
7.1 General structures of the Higgs branch Lagrangian
7.2 Hypermultiplets revisited
7.3 The hyperkähler quotient
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
9.3 Weak-coupling limit and trifundamentals
9.4 Strong-coupling limit
9.5 Generalization
9.6 Theories with less ﬂavors revisited
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
10.5 More general Argyres-Douglas CFTs:

X_{N}

Y_{N}

11 Theories with other simple gauge groups
11.1 Semiclassical analysis
11.2 Pure

S U (N)

theory
11.3

S U (N)

theory with fundamental ﬂavors
11.4

S O (2 N)

theories
11.5 Argyres-Douglas CFTs
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)

N_{f} = 2 N

ﬂavors, part I
12.2

S U (N)

quiver theories and tame punctures
12.3 S-dual of

S U (N)

N_{f} = 2 N

ﬂavors, part II
12.4 Applications
12.5 Tame punctures and Higgsing
13 Conclusions and further directions
References