Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Seiberg-Witten Geometries Revisited

Yuji Tachikawa, Seiji Terashima

TL;DR

The paper establishes a universal Seiberg-Witten geometry for 4d $\mathcal{N}=2$ theories with a single simply-laced gauge group $G$ and matter in representations with $b_R\le h^\vee$, realized by a non-compact Calabi-Yau defined from $W_G$ and $X_R$ as $z + \frac{\Lambda^{2h^\vee}}{z} \prod_R \Lambda^{-b_R} X_R = W_G$. It systematically interprets $X_R$ as encoding hypermultiplets and demonstrates the construction's consistency with 6d $(2,0)$ compactifications on spheres with punctures, using decoupling to generate $X_R$ for exceptional representations (notably the $\mathbf{56}$ of $E_7$). The paper provides detailed case studies—the $2$-index antisymmetric of $SU(n)$, the $\mathbf{20}$ of $SU(6)$, and the $\mathbf{56}$ of $E_7$—to illustrate the method, including explicit Calabi-Yau equations, discriminant analyses, and Hitchin-field puncture data. These results unify multiple previous SW solutions within a single geometric framework and point to extensions to non-simply-laced groups and broader representation sets, highlighting the deep link between Calabi-Yau geometry, 6d defects, and 4d dynamics.

Abstract

We provide a uniform solution to 4d N=2 gauge theory with a single gauge group G=A,D,E when the one-loop contribution to the beta function from any irreducible component R of the hypermultiplets is less than or equal to half of that of the adjoint representation. The solution is given by a non-compact Calabi-Yau geometry, whose defining equation is built from explicitly known polynomials W_G and X_R, associated respectively to the gauge group G and each irreducible component R. We provide many pieces of supporting evidence, for example by analyzing the system from the point of view of the 6d N=(2,0) theory compactified on a sphere.

Seiberg-Witten Geometries Revisited

TL;DR

The paper establishes a universal Seiberg-Witten geometry for 4d theories with a single simply-laced gauge group and matter in representations with , realized by a non-compact Calabi-Yau defined from and as . It systematically interprets as encoding hypermultiplets and demonstrates the construction's consistency with 6d compactifications on spheres with punctures, using decoupling to generate for exceptional representations (notably the of ). The paper provides detailed case studies—the -index antisymmetric of , the of , and the of —to illustrate the method, including explicit Calabi-Yau equations, discriminant analyses, and Hitchin-field puncture data. These results unify multiple previous SW solutions within a single geometric framework and point to extensions to non-simply-laced groups and broader representation sets, highlighting the deep link between Calabi-Yau geometry, 6d defects, and 4d dynamics.

Abstract

We provide a uniform solution to 4d N=2 gauge theory with a single gauge group G=A,D,E when the one-loop contribution to the beta function from any irreducible component R of the hypermultiplets is less than or equal to half of that of the adjoint representation. The solution is given by a non-compact Calabi-Yau geometry, whose defining equation is built from explicitly known polynomials W_G and X_R, associated respectively to the gauge group G and each irreducible component R. We provide many pieces of supporting evidence, for example by analyzing the system from the point of view of the 6d N=(2,0) theory compactified on a sphere.

Paper Structure

This paper contains 27 sections, 110 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Generalities
  3. Motivating examples
  4. $\mathcal{N}=2$ gauge theory without matters
  5. $\mathrm{SU}(n)$ with fundamental hypermultiplets
  6. Hypermultiplet geometries
  7. General recipe
  8. Case studies
  9. 2-index antisymmetric tensor of $\mathrm{SU}(n)$
  10. Derivation
  11. Confirming the discriminant
  12. 3-index antisymmetric tensor $\mathbf{20}$ of $\mathrm{SU}(6)$
  13. Decoupling from $\mathbf{32}$ of $\mathrm{SO}(12)$
  14. Comparison with 6d $\mathcal{N}=(2,0)$ theory: introductory remarks
  15. Half $\mathbf{20}$ and three fundamentals
  16. ...and 12 more sections

Figures (2)

  • Figure 1: a) The base of the Seiberg-Witten geometry in the weakly-coupled region. b) a half of the Seiberg-Witten geometry. This can be though of as the geometry representing free hypermultiplets.
  • Figure 2: The same non-compact Calabi-Yau geometry \ref{['same']} leads to two completely different sets of codimension-two "punctures" as 6d $\mathcal{N}=(2,0)$ theory on a sphere. On the left, we chose the projection to the sphere parameterized by $z'$, \ref{["z'"]}. The pair of a simple puncture and a full puncture gives us six fundamentals, and the pair of two punctures with $\mathbb{Z}_2$ branch cuts produces one full hypermultiplet in $\mathbf{20}$. On the right, the projection was to the sphere parameterized by $z"$, \ref{['z"']}. Two pairs of a puncture of type $[4,2]$ and another of type $[2,2,2]$ each give one half-hypermultiplet in $\mathbf{20}$ and three fundamentals.