On integrable systems and supersymmetric gauge theories

Abstract

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 possible the parallels between standard quantum field theory results and solutions to integrable systems are discussed.

String theory or field theory?

“…For example, in the Seiberg-Witten theory now there exists a reasonable answer only to the first question 27 , and it is very important that the Wilsonian effective action in the massless sector can be expressed (4.19) via a function of several complex variables. Thus, it is the knowledge of the function F as function of moduli and all its derivatives, say the expansion over the sources T which gives the most complete information about the theory.…”

“…Equation det(λ − Φ(z)) = 0 (literally corresponding to the theory with broken N = 4 supersymmetry) in the limit m → ∞ and τ → +i∞ with m N exp(iπτ ) = Λ N turns exactly into eq. (4.11), the details and references can be found in [8,27].…”

“…whose properties ensure (see the details, say, in [8,27]), that the period matrix of the Riemann surface (4.11) can be expressed in terms of the derivatives…”

String theory or field theory?

“…It 27) where g str = exp(Φ) . This relation will be essentially used below in discussion of the nonperturbative string theory.…”

Seiberg-Witten Theory and Calogero-Moser Systems*

“…The monodromies and instanton corrections in the corresponding prepotential were determined in [31]. More recently, Seiberg-Witten curves for many other theories have been found, based on integrable models [12][13][14], MTheory [15], and geometric engineering [16]. Monodromies and instanton corrections have been determined in several important cases, including for the SU (N ) gauge theory, with hypermultiplet in the symmetric or anti-symmetric representation [17].…”