On N=2 SUSY gauge theories and integrable systems

The properties of the N=2 SUSY gauge theories underlying the Seiberg-Witten hypothesis are discussed. The main ingredients of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential are presented, the invariant sense of these definitions is illustrated. Recently found exact nonperturbative solutions to N=2 SUSY gauge theories are formulated using the methods of the theory of integrable systems and where poss...

This is an announcement of some of the results of a longer paper where the supersymmetric vacua of two dimensional N=2 susy gauge theories with matter are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The correspondence between the Heisenberg spin chain and the two dimensional U(N) theory with fundamental hypermultiplets is reviewed in detail. We demonstrate the isomorphism of the equivariant quantum cohomology of the cot...

A summary of results is presented, which provide exact description of the low-energy $4d$ $N=2$ and $N=4$ SUSY gauge theories in terms of $1d$ integrable systems.

Some additional references are included on the last 3 pages.

This is a survey of the work of Seiberg and Witten on 4-dimensional N=2 supersymmetric Yang-Mills theory and of some of its recent extensions, written for mathematicians. The point of view is that of algebraic geometry and integrable systems. An introductory chapter tries to describe some of the relevant physics for a reader with no physics background. This is followed by a review of the relevant properties of integrable systems. The remaining chapters describe the specific i...

These lectures are devoted to the low energy limit of \N2 SUSY gauge theories, which is described in terms of integrable systems. A special emphasis is on a duality that naturally acts on these integrable systems. The duality turns out to be an effective tool in constructing the double elliptic integrable system which describes the six-dimensional Seiberg-Witten theory. At the same time, it implies a series of relations between other Seiberg-Witten systems.

This is the writeup of the lectures given at the Winter School "YRISW 2018" to appear in a special issue of JPhysA. In the first part of these lecture notes we review some important facts about 4D $\mathcal{N}=2$ SCFTs. We begin with basic textbook material, the supersymmetry algebra and its massless representations and the construction of Lagrangians using superspace. Then we turn to more modern topics, the study of the $\mathcal{N}=2$ SCA and its representation theory. Our ...

These are expanded notes of lectures given at the summer school "Gif 2000" in Paris. They constitute the first part of an "Introduction to supersymmetry and supergravity" with the second part on supergravity by J.-P. Derendinger to appear soon. The present introduction is elementary and pragmatic. I discuss: spinors and the Poincar\'e group, the susy algebra and susy multiplets, superfields and susy lagrangians, susy gauge theories, spontaneously broken susy, the non-linear s...

In these notes we attempt to give a pedagogical introduction to the work of Seiberg and Witten on S-duality and the exact results of N=2 supersymmetric gauge theories with and without matter. The first half is devoted to a review of monopoles in gauge theories and the construction of supersymmetric gauge theories. In the second half, we describe the work of Seiberg and Witten.

This is the introduction to the collection of review articles "Exact results on N=2 supersymmetric gauge theories". The first three sections are intended to give a general overview over the physical motivations behind this direction of research, and some of the developments that initiated this project. These sections are written for a broad audience of readers with interest in quantum field theory, assuming only very basic knowledge of supersymmetric gauge theories and string...