Seiberg-Witten geometry of mass deformed N = 2 superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space M of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space Bun G (E) of holomorphic G C -bundles on a (possibly degenerate) elliptic curve E defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group G. The integrable systems P underlying, or, rather, overlooking the special geometry of M are identified. The moduli spaces of framed G-instantons on R 2 × T 2 , of G-monopoles with singularities on R 2 × S 1 , the Hitchin systems on curves with punctures, as well as various spin chains play an important rôle in our story. We also comment on the higher dimensional theories. In the companion paper the quantum integrable systems and their connections to the representation theory of quantum affine algebras will be discussed.
ContentsAcknowledgments iii Chapter 1. Introduction 1.1. Organization of the material 1.2. Notations Chapter 2. Supersymmetric quiver theories 2.1. Gauge group, matter fields, couplings, parameters Chapter 3. The ADE classifications of superconformal N = 2 theories 3.1. Beta functions and Cartan matrix 3.2. Class I theories 3.3. Class II theories 3.4. Class II* theories Chapter 4. Low-energy effective theory 4.1. Special Kähler geometry 4.2. Extended moduli space 4.3. Finite size effects 4.4. The appearance of an integrable system 4.5. Integrable systems from classical gauge theories 4.6. Extended moduli space as a complex integrable system Chapter 5. The limit shape equations 5.1. The amplitude functions 5.2. The limit shape prepotential 5.3. The iWeyl group. 5.4. Moduli of vacua and mass parameters Chapter 6. Solution of the limit shape equations 6.1. From invariants to the curve 6.2. The cameral curve as a modular object 6.3. Spectral curves 6.4. Obscured curve iv CONTENTS v Chapter 7. The Seiberg-Witten curves in some detail 7.1. Class I theories of A type 7.2. Class I theories of D type 7.3. Class I theories of E type 7.4. Class II theories of A type and class II* theories 7.5. Class II theories of D type 7.6. Class II theories of E type Chapter 8. The integrable systems of monopoles and instantons 8.1. Periodic Monopoles and the phase space of Class I theories105 8.2. Double-periodic instantons and Class II theories 8.3. Noncommutative instantons and Class II* theories Chapter 9. Higher dimensional theories Appendix A.