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Global forms of $\mathcal{N}=4$ theories and non-minimal Seiberg-Witten solutions

Robert Moscrop

TL;DR

This work recasts the refinement of BPS charge lattices into fibre-wise isogenies of polarised abelian varieties, enabling a unified ACI-system description of absolute $\mathcal{N}=4$ theories. By computing explicit relative and absolute fibres for several gauge algebras, it demonstrates how period matrices transform under fibre isogenies and how the defect group governs allowed line-operator refinements. It shows that many absolute $\mathcal{N}=4$ theories do not admit Seiberg-Witten curves of minimal genus, due to the Schottky problem and Hurwitz bounds on automorphisms, highlighting limitations of Jacobian-based SW descriptions in higher rank. The results illuminate the residual $S$-duality structure via stabilisers and Hecke-type actions on the moduli of abelian fibres and suggest Prym/Donagi constructions as a route to higher-genus SW frameworks for these theories.

Abstract

To each four dimensional $\mathcal{N}\geq 2$ supersymmetric quantum field theory, one can associate an algebraic completely integrable (ACI) system that encodes the low energy dynamics of theory. In this paper we explicitly derive the appropriate ACI systems for the global forms of $\mathcal{N}=4$ super Yang-Mills (sYM) using isogenies of polarised abelian varieties. In doing so, we relate the complex moduli of the resulting varieties to the exactly marginal coupling of the theory, thus allowing us to probe the $S$-duality groups of the global forms. Finally, we comment on whether the resulting varieties are the Jacobians of a minimal genus Riemann surface, coming to the conclusion that many global forms of $\mathcal{N}=4$ sYM do not admit a minimal genus Seiberg-Witten curve that correctly reproduces the global form.

Global forms of $\mathcal{N}=4$ theories and non-minimal Seiberg-Witten solutions

TL;DR

This work recasts the refinement of BPS charge lattices into fibre-wise isogenies of polarised abelian varieties, enabling a unified ACI-system description of absolute theories. By computing explicit relative and absolute fibres for several gauge algebras, it demonstrates how period matrices transform under fibre isogenies and how the defect group governs allowed line-operator refinements. It shows that many absolute theories do not admit Seiberg-Witten curves of minimal genus, due to the Schottky problem and Hurwitz bounds on automorphisms, highlighting limitations of Jacobian-based SW descriptions in higher rank. The results illuminate the residual -duality structure via stabilisers and Hecke-type actions on the moduli of abelian fibres and suggest Prym/Donagi constructions as a route to higher-genus SW frameworks for these theories.

Abstract

To each four dimensional supersymmetric quantum field theory, one can associate an algebraic completely integrable (ACI) system that encodes the low energy dynamics of theory. In this paper we explicitly derive the appropriate ACI systems for the global forms of super Yang-Mills (sYM) using isogenies of polarised abelian varieties. In doing so, we relate the complex moduli of the resulting varieties to the exactly marginal coupling of the theory, thus allowing us to probe the -duality groups of the global forms. Finally, we comment on whether the resulting varieties are the Jacobians of a minimal genus Riemann surface, coming to the conclusion that many global forms of sYM do not admit a minimal genus Seiberg-Witten curve that correctly reproduces the global form.

Paper Structure

This paper contains 23 sections, 4 theorems, 95 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complex tori and isogenies
  4. Polarisations
  5. Jacobians and the Schottky problem
  6. ACI systems for absolute N=4 theories
  7. Refinements and isogenies
  8. Fibres for N=4 theories
  9. The relative fibres
  10. The pure refinements
  11. Absolute fibres for A-type sYM
  12. Non-simply laced example
  13. S-duality action on fibres
  14. Relative conformal manifolds
  15. Absolute S-duality groups
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let $\mathcal{N}$ be a polarisation on $X$. Given an isogeny $f:X\rightarrow X'$, there exists a polarisation $\mathcal{L}$ on $X'$ satisfying $\mathcal{N}=f^*\mathcal{L}$ if and only if $\ker f$ is an isotropic subgroup of $K(\mathcal{N})$ with respect to the pairing $\mathrm{e}^\mathcal{N}$.

Figures (3)

  • Figure 1: Left: The tiling of the Poincaré disk by the Schwarz triangle with internal angles $\pi/2$, $\pi/3$ and $\pi/7$. Right: The order-$7$ triangular tiling of the Poincaré disk. Every Hurwitz surface admits a triangulation which is a quotient of this tiling. This image was generated with the Python code available at sch_py.
  • Figure 2: Left: The root lattice $\Gamma_r\cong\langle\alpha_1,\alpha_2\rangle_\mathbb{Z}$ of $A_2$. Right: A refinement of $\Gamma_r$ given by the weight lattice $\Gamma_w \cong \langle \alpha_1,\omega\rangle_\mathbb{Z}$. Highlighted are the nodes of $\Gamma_r$ present in $\Gamma_w$.
  • Figure 3: A fundamental domain $\mathcal{D}$ for the fractional linear action of $\mathrm{SL}_2(\mathbb{Z})$ on the upper half plane. Shown are some images of $\mathcal{D}$ and the stacky points which have are fixed under some subgroup of $\mathrm{SL}_2(\mathbb{Z})$.

Theorems & Definitions (16)

  • Definition 2.1
  • Lemma 2.1
  • Theorem 2.2
  • Corollary 2.1
  • Theorem 2.3
  • Definition 3.1
  • Remark
  • Example 3.1
  • Remark
  • Example 3.2
  • ...and 6 more