[Paper Review] Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories

Nikita Nekrasov, Vasily Pestun|arXiv (Cornell University)|Nov 12, 2012

Black Holes and Theoretical PhysicsPhysics and Astronomy152 references139 citations

TL;DR

This paper determines the Seiberg-Witten geometry of four-dimensional N=2 quiver gauge theories with mass deformations, identifying the vacuum moduli space M as the moduli space of genus-zero holomorphic (quasi)maps to the moduli space of holomorphic G-bundles on a degenerate elliptic curve, where G is the ADE gauge group. The underlying integrable systems are identified, linking instantons, monopoles, Hitchin systems, and spin chains to the special geometry of M.

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 of holomorphic G-bundles on a (possibly degenerate) elliptic curve defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group G. The integrable systems underlying, or, rather, overlooking the special geometry of M are identified. The moduli spaces of framed G-instantons on R^2xT^2, of G-monopoles with singularities on R^2xS^1, the Hitchin systems on curves with punctures, as well as various spin chains play an important role 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

Motivation & Objective

• To determine the Seiberg-Witten geometry of mass-deformed N=2 superconformal ADE quiver gauge theories in four dimensions.
• To identify the vacuum moduli space M as the moduli space of genus-zero holomorphic (quasi)maps to the moduli space of holomorphic G-bundles on a possibly degenerate elliptic curve.
• To uncover the integrable systems underlying the special geometry of M, linking gauge theory to geometric and integrable structures.
• To establish connections between the vacuum geometry and physical systems such as framed G-instantons on R²×T², singular G-monopoles on R²×S¹, and Hitchin systems with punctures.
• To lay the groundwork for understanding quantum integrable systems and their relation to quantum affine algebras, as explored in the companion paper.

Proposed method

• Solving the limit shape equations derived from the N=2 gauge theory to determine the vacuum moduli space M.
• Using the microscopic gauge couplings to define the elliptic curve and the moduli space of holomorphic G-bundles on it.
• Analyzing genus-zero holomorphic (quasi)maps into the moduli space of G-bundles to characterize M.
• Identifying the integrable systems that underlie or overlook the special geometry of M through geometric and gauge-theoretic constructions.
• Employing the framework of framed G-instantons on R²×T² and G-monopoles with singularities on R²×S¹ as key physical realizations.
• Applying Hitchin systems on punctured curves and relating them to the vacuum geometry via geometric engineering.

Experimental results

Research questions

• RQ1 How is the vacuum moduli space M of mass-deformed N=2 ADE quiver gauge theories geometrically characterized?
• RQ2 What is the role of the elliptic curve and its moduli space of holomorphic G-bundles in determining the Seiberg-Witten geometry?
• RQ3 Which integrable systems underlie or are associated with the special geometry of M?
• RQ4 How do framed G-instantons on R²×T² and singular G-monopoles on R²×S¹ contribute to the geometric structure of M?
• RQ5 What is the connection between the vacuum geometry and Hitchin systems on punctured Riemann surfaces?

Key findings

• The vacuum moduli space M is identified as the moduli space of genus-zero holomorphic (quasi)maps to the moduli space of holomorphic G-bundles on a possibly degenerate elliptic curve.
• The Seiberg-Witten geometry of the quiver gauge theory is fully determined by the geometric data of the elliptic curve and the G-bundle moduli space, encoded via the microscopic gauge couplings.
• The integrable systems underlying the special geometry of M are explicitly identified, linking the gauge theory to integrable structures in mathematical physics.
• Framed G-instantons on R²×T² and G-monopoles with singularities on R²×S¹ are shown to play a central role in realizing the vacuum geometry.
• Hitchin systems on curves with punctures emerge naturally as part of the geometric description of M, providing a bridge to algebraic geometry.
• The framework sets the stage for the quantum lift of these integrable systems and their connection to the representation theory of quantum affine algebras, as detailed in the companion paper.