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Solution of N=2 Gauge Theories via Compactification to Three Dimensions

Anton Kapustin

TL;DR

This work develops a novel, exact approach to solving finite N=2 gauge theories realized by brane configurations. By compactifying to three dimensions and applying a 3d mirror-like transform, the four-dimensional Coulomb branch is mapped to the Higgs branch of a five-dimensional magnetic theory with three-dimensional impurities, which is uncorrected by quantum effects and captured by a Hitchin-type integrable system on a punctured torus. In the decompactification limit, the construction yields the Seiberg–Witten spectral curve, matching the M-theory (Witten) description and clarifying the role of mass deformations and U(1) factor freezing. The framework unifies Hitchin and Donagi–Markman perspectives in elliptic models and suggests avenues for extension to non-elliptic and asymptotically free theories, with broad implications for exact nonperturbative dynamics of N=2 theories.

Abstract

A number of N=2 gauge theories can be realized by brane configurations in Type IIA string theory. One way of solving them involves lifting the brane configuration to M-theory. In this paper we present an alternative way of analyzing a subclass of these theories (elliptic models). We observe that upon compactification on a circle one can use a version of mirror symmetry to map the original brane configuration into one containing only D-branes. Simultaneously the Coulomb branch of the four-dimensional theory is mapped to the Higgs branch of a five-dimensional theory with three-dimensional impurities. The latter does not receive quantum corrections and can be analyzed exactly. The solution is naturally formulated in terms of an integrable system, which is a version of a Hitchin system on a punctured torus.

Solution of N=2 Gauge Theories via Compactification to Three Dimensions

TL;DR

This work develops a novel, exact approach to solving finite N=2 gauge theories realized by brane configurations. By compactifying to three dimensions and applying a 3d mirror-like transform, the four-dimensional Coulomb branch is mapped to the Higgs branch of a five-dimensional magnetic theory with three-dimensional impurities, which is uncorrected by quantum effects and captured by a Hitchin-type integrable system on a punctured torus. In the decompactification limit, the construction yields the Seiberg–Witten spectral curve, matching the M-theory (Witten) description and clarifying the role of mass deformations and U(1) factor freezing. The framework unifies Hitchin and Donagi–Markman perspectives in elliptic models and suggests avenues for extension to non-elliptic and asymptotically free theories, with broad implications for exact nonperturbative dynamics of N=2 theories.

Abstract

A number of N=2 gauge theories can be realized by brane configurations in Type IIA string theory. One way of solving them involves lifting the brane configuration to M-theory. In this paper we present an alternative way of analyzing a subclass of these theories (elliptic models). We observe that upon compactification on a circle one can use a version of mirror symmetry to map the original brane configuration into one containing only D-branes. Simultaneously the Coulomb branch of the four-dimensional theory is mapped to the Higgs branch of a five-dimensional theory with three-dimensional impurities. The latter does not receive quantum corrections and can be analyzed exactly. The solution is naturally formulated in terms of an integrable system, which is a version of a Hitchin system on a punctured torus.

Paper Structure

This paper contains 8 sections, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. N=2 Gauge Theories and Integrable Models
  3. Compactification to Three Dimensions and the Mirror Transform
  4. Solution of the Models
  5. The Higgs Branch of the Impurity Theory
  6. The Decompactification Limit
  7. Comparison with M-theory Curves
  8. Discussion

Figures (2)

  • Figure 1: After $x^3$ is reinterpreted as the M-theory circle, the "electric" D4-branes become D4-branes wrapped around $x^6$ and $x^{10}$, while NS5-branes become D4$'$-branes localized in the $x^6,x^{10}$ directions. Here the positions of the D4$'$-branes are shown as punctures on the $T^2$ parametrized by $x^6,x^{10}$.
  • Figure 2: The Wilson line around the puncture created by the D4$'$-brane can be deformed to a product of the Wilson lines along the two contours shown in the figure.