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Introduction to Seiberg-Witten Theory and its Stringy Origin

W. Lerche

TL;DR

The paper provides an accessible entry into the Seiberg–Witten solution of $N=2$ supersymmetric Yang–Mills theory, highlighting how the exact low-energy dynamics are encoded in a holomorphic prepotential $\mathcal{F}(a)$ and a multi-valued moduli space shaped by monodromy. It develops the two-region structure for SU(2) with a dual pair $(a,a_D)$ and derives the exact quantum moduli space, the BPS spectrum, and the Picard–Lefschetz–type monodromy data, all governed by elliptic curves and their period integrals. It then explains how these results can be re-derived from heterotic–type II string duality via ALE fibrations and Calabi–Yau geometry, linking spectral curves to non-perturbative string physics. The analysis generalizes to ADE groups and to stringy realizations, establishing a deep connection between vanishing cycles, monodromies, and non-perturbative field theory data, with broad implications for dualities and quantum geometry.

Abstract

We give an elementary introduction to the recent solution of $N=2$ supersymmetric Yang-Mills theory. In addition, we review how it can be re-derived from string duality.

Introduction to Seiberg-Witten Theory and its Stringy Origin

TL;DR

The paper provides an accessible entry into the Seiberg–Witten solution of supersymmetric Yang–Mills theory, highlighting how the exact low-energy dynamics are encoded in a holomorphic prepotential and a multi-valued moduli space shaped by monodromy. It develops the two-region structure for SU(2) with a dual pair and derives the exact quantum moduli space, the BPS spectrum, and the Picard–Lefschetz–type monodromy data, all governed by elliptic curves and their period integrals. It then explains how these results can be re-derived from heterotic–type II string duality via ALE fibrations and Calabi–Yau geometry, linking spectral curves to non-perturbative string physics. The analysis generalizes to ADE groups and to stringy realizations, establishing a deep connection between vanishing cycles, monodromies, and non-perturbative field theory data, with broad implications for dualities and quantum geometry.

Abstract

We give an elementary introduction to the recent solution of supersymmetric Yang-Mills theory. In addition, we review how it can be re-derived from string duality.

Paper Structure

This paper contains 8 sections, 59 equations, 8 figures.

Table of Contents

  1. Introduction
  2. $N=2$ Yang-Mills Theory
  3. Overview
  4. The Semi-Classical Theory for $G=SU(2)$
  5. The exact quantum moduli space
  6. Solving the monodromy problem
  7. The BPS Spectrum
  8. Picard-Fuchs equations

Figures (8)

  • Figure 1: At scales above the Higgs VEV $a$, the masses of the non-abelian gauge bosons, $W^\pm$, are negligible, and we can see the ordinary running of the coupling constant of an asymptotically free theory. At scales below $a$, $W^\pm$ freeze out, and we are left with just an effective $U(1)$ gauge theory with vanishing $\beta$-function.
  • Figure 2: The transition from the classical to the exact quantum theory involves splitting and shifting of the strong coupling singularity away from $u=0$ to $u=\pm \Lambda^2$.
  • Figure 3: The exact quantum moduli space is covered by three distinct regions, in the center of each of which the theory is weakly coupled when choosing suitable local variables. A local effective lagrangian exists in each coordinate patch, representing a particular perturbative approximation. None of such lagrangians is more fundamental than the other ones, and no local lagrangian exists that would be globally valid throughout the moduli space.
  • Figure 4: Monodromy paths in the $u$-plane.
  • Figure 5: Basis of one-cycles on the torus.
  • ...and 3 more figures