Texts and Readings in Physical Sciences

Understanding the dynamics of gauge theories is important because all known interactions are based on the principle of local gauge symmetry. Beyond the perturbative regime however, this is a notoriously difficult problem. Demanding invariance under supersymmetry turns out to provide a better handle, and allows one to analyze supersymmetric gauge theories over a larger region in the space of parameters. Quantum field theories in four dimensions with extended N=2 supersymmetry are further constrained and have, therefore, been a fertile field of research in theoretical physics for quite some time. Moreover, there are mathematical ramifications that have lead to a successful dialogue with differential and algebraic geometry.

These lecture notes aim to introduce the fascinating developments in the understanding of N=2 supersymmetric gauge theories in a coherent fashion to a serious student of modern theoretical physics. Starting with a gentle introduction to electric-magnetic duality, the author guides the reader along the important milestones in the subject among which are the work of Seiberg and Witten, Nekrasov, Gaiotto and many others. This is an advanced graduate level text, and it assumes that the reader has a working knowledge of supersymmetry, including the formalism of superfields as well as techniques of quantum field theory such as regularization, renormalization and anomalies.

After his graduation from the University of Tokyo, Yuji Tachikawa worked at the Institute for Advanced Study, Princeton and the Kavli Institute for Physics and Mathematics of the Universe, Kashiwa. Presently at the Department of Physics, University of Tokyo, Tachikawa is the author of several important papers in supersymmetric quantum field theories and string theory.
1 Electromagnetic Duality and Monopoles
2 N=2 Multiplets and Lagrangians
3 Renormalization and Anomaly
4 Seiberg–Witten Solution to Pure SU.2/ Theory
5 SU.2/ Theory with One Flavor
6 Curves and 6d N=(2, 0) Theory
7 Higgs Branches and Hyperkähler Manifolds
8 SU.2/ Theory with 2 and 3 Flavors
9 SU.2/ Theory with Four Flavors and Gaiotto's Duality
10 Argyres–Douglas CFTs
11 Theories with Other Simple Gauge Groups
12 Argyres–Seiberg–Gaiotto Duality for SU.N/ Theory
13 Conclusions and Further Directions
A Prepotential and the Instanton Computation
B The Zoo of N=2 Theories