On N=2 SUSY gauge theories and integrable systems

We summarize recent results on the resolution of two intimately related problems, one physical, the other mathematical. The first deals with the resolution of the non-perturbative low energy dynamics of certain N=2 supersymmetric Yang-Mills theories. We concentrate on the theories with one massive hypermultiplet in the adjoint representation of an arbitrary gauge algebra G. The second deals with the construction of Lax pairs with spectral parameter for certain classical mecha...

We give an elementary introduction to the recent solution of $N=2$ supersymmetric Yang-Mills theory. In addition, we review how it can be re-derived from string duality.

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. This is a write-up of the author's lectures at Tohoku University, Nagoya University and Rikkyo University. Comments will be appreciated.

We obtain the Seiberg-Witten geometry for four-dimensional N=2 gauge theory with gauge group SO(2N_c) (N_c \leq 5) with massive spinor and vector hypermultiplets by considering the gauge symmetry breaking in the N=2 $E_6$ theory with massive fundamental hypermultiplets. In a similar way the Seiberg-Witten geometry is determined for N=2 SU(N_c) (N_c \leq 6) gauge theory with massive antisymmetric and fundamental hypermultiplets. Whenever possible we compare our results express...

We describe how the ingredients and results of the Seiberg-Witten solution to N=2 supersymmetric U(N) gauge theory may be obtained from a matrix model.

The Coulomb branch of $N=2$ supersymmetric gauge theories in four dimensions is described in general by an integrable Hamiltonian system in the holomorphic sense. A natural construction of such systems comes from two-dimensional gauge theory and spectral curves. Starting from this point of view, we propose an integrable system relevant to the $N=2$ $SU(n)$ gauge theory with a hypermultiplet in the adjoint representation, and offer much evidence that it is correct. The model h...

We classify all possible four-dimensional N=2 supersymmetric UV-complete gauge theories composed of semi-simple gauge groups and hypermultiplets. We also give appropriate references for all theories with known Seiberg-Witten solutions.

A new method to obtain the Picard-Fuchs equations of effective, N=2 supersymmetric gauge theories with massive matter hypermultiplets in the fundamental representation is presented. It generalises a previously described method to derive the Picard-Fuchs equations of both pure super Yang-Mills and supersymmetric gauge theories with massless matter hypermultiplets. The techniques developed are well suited to symbolic computer calculations.

The Seiberg-Witten solution plays a central role in the study of N=2 supersymmetric gauge theories. As such, it provides a proving ground for a wide variety of techniques to treat such problems. In this review we concentrate on the role of IIA string theory/M theory and the Dijkgraaf-Vafa matrix model, though integrable models and microscopic instanton calculations are also of considerable importance in this subject.

The integrable model corresponding to the ${\cal N}=2$ supersymmetric SU(N) gauge theory with matter in the symmetric representation is constructed. It is a spin chain model, whose key feature is a new twisted monodromy condition.