IPMU 13-0234
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


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 SU(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.

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 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
 4.4 Less supersymmetric cases
 4.5 SU(2) vs SO(3)
5 SU(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 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
 9.3 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 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
 10.5 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
 11.3 SU(N) theory with fundamental flavors
 11.4 SO(2N) theories
 11.5 Argyres-Douglas CFTs
 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
 12.2 SU(N) quiver theories and tame punctures
 12.3 S-dual of SU(N) with Nf = 2N flavors, part II
 12.4 Applications
 12.5 Tame punctures and Higgsing
13 Conclusions and further directions