Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories

Abstract

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.

Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations

“…In particular, a natural conjecture is that the proper surface defect arrangement in the quiver gauge theory based on a quiver of ADE or Â, D, Ê-type, the sl 2 spin chain would be replaced by the corresponding spin chain based on the Yangian of the corresponding Lie algebra. In the quasiclassical limit this is supported by the identification [21] of Seiberg-Witten geometries of these theories with the moduli spaces of ADE monopoles on R 2 × S 1 or instantons on R 2 × T 2 . The deformation quantization of these spaces produces the corresponding Yangian algebras [60,82].…”

“…(2.14) This is the partition function of the rank N A 1 theory [21], i.e. U (N ) gauge theory with 2N fundamentals.…”

“…With the same restriction on the spin representations, the IR duality discovered in [27] between four-dimensional N = 2 theories and two-dimensional N = (2, 2) theories gave a four-dimensional account for the quantum spin chains. Finally, it was shown in [42] that the classical XXX sl 2 spin chain arises in the Seiberg-Witten geometry of the four-dimensional N = 2 theory [21,60], as well as a relation between the XXX sl 2 spin chain coordinate systems and the defect gauge theory parameters, with more general sl 2 -representations which are neither highest-weight nor lowest-weight. The extension to the supergroup gauge theories is also discussed in [66][67][68].…”

“…Indeed, for S-class N = 2 theories [16] the associated integrable systems can be argued to be precisely Hitchin integrable systems [17]. Another vast class of theories, namely the ones corresponding to the quivers with unitary gauge groups, are demonstrated [21] to be dual to algebraic integrable systems on the moduli spaces of double-periodic instantons or periodic monopoles. The classical spin chains are connected through the equivalence of the spectral curves [22,23].…”

Large deviations for discrete $β$-ensembles

“…In particular, the rightmost particle converges to the β-Tracy-Widom distribution. The way these results were established in [GH19] is through an intricate use of what are called discrete loop equations or Nekrasov's equations [BGG17,Nek16,NP12,NPS13]. Compared to [GH19], the present paper does not utilize loop equations and relies on more direct combinatorial constructions and estimates.…”