13 Conclusions and further directions

In this lecture note, we first discussed the Lagrangian of 𝒩=2 supersymmetric gauge theory, and then studied the Coulomb and Higgs branches of SU(2) gauge theories with various number of flavors. Two related concepts, the Seiberg-Witten curve and the ultraviolet curve played very important roles along the way. We then analyzed what happens when Coulomb branch vevs or exactly-marginal coupling parameters are finely tuned. Sometimes the limit was described by a dual weakly-coupled gauge theory, as was the case with SU(2) theory with four flavors. Most often, however, we saw that we end up with new superconformal field theories, of Argyres-Douglas-type or of Gaiotto-type.

For example, we saw the theories ADNf=1,2,3(SU(2)) and MN(E6,7,8) in Sec. 10.4, the theories XN and Y N in Sec. 10.5, RN in Sec. 12.3 and TN in Sec. 12.4. More and more 𝒩=2 superconformal theories are being discovered, see e.g. [65]. This means that, to fully understand the interrelations of 𝒩=2 supersymmetric systems, we cannot restrict our attention to just 𝒩=2 theories composed of vector multiplets and hypermultiplets.

The topics we covered in this lecture note are only a tip of a huge iceberg that is the study of 𝒩=2 dynamics, and there are many other further directions of research. Let us list some of them.17 First, we can put an 𝒩=2 theory on a nontrival manifold:

Second, we can study dynamical excitations and externally-introduced operators of these theories:

On these topics, the review [101] is a great source of information, although the review itself is meant for mathematicians.

Third, the method described in this lecture note is not yet powerful enough to solve arbitrary 𝒩=2 gauge theories. Many 4d 𝒩=2 theories do come from the 6d 𝒩=(2, 0) theory, but there are also many which presently do not. Therefore we should also study alternative approaches.

Fourth, there are many properties of 𝒩=2 theories which are satisfied by all known examples, but we do not currently have any way to derive them. It would be fruitful to devise new methods to study these properties. Let us list a few questions in this direction.

Finally, the author would like to emphasize that even such innocent looking gauge theories as

have not been solved yet. He would be happy to offer a dinner at the Sushi restaurant in the Kashiwa campus to the first person who finds the solution to either of the two theories. There are many other 𝒩=2 gauge theories without known solutions, as listed in [107]. So this field should be considered still wide-open.

Hopefully, those readers who came to this point should be at least moderately equipped to tackle these and other recent articles on 𝒩=2 supersymmetric theories. It would be a pleasure for the author if they would continue the study and contribute to extend the frontier of the research.