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Special Arithmetic of Flavor

Matteo Caorsi, Sergio Cecotti

TL;DR

The work reframes the rank-1 ${\cal N}=2$ SW landscape through Diophantine geometry by treating special geometries as elliptic surfaces over the chiral ring, using the Kodaira-Néron model to classify non-free theories via the pair $(\mathcal{E},F_\infty)$. It shows that SW completeness aligns with the safely irrelevant conjecture and that the flavor content is encoded in the Mordell-Weil lattice, with $E_8$-root curves yielding flavor roots through good-position integral/narrow sections; the Oguiso-Shioda tables then determine admissible flavor groups. The analysis connects discrete gaugings to base changes of elliptic surfaces, and provides a comprehensive, geometry-driven account that agrees with Argyres et al. across known rank-1 classifications. The framework unifies UV behavior, flavor symmetry, and deformation data into a coherent arithmetic picture, enabling systematic exploration of discrete gauging and flavor-root structures via rational elliptic surfaces and their Mordell-Weil lattices.

Abstract

We revisit the classification of rank-1 4d $\mathcal{N}=2$ QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-Néron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs $(\mathcal{E},F_\infty)$ where $\mathcal{E}$ is a relatively minimal, rational elliptic surface with section, and $F_\infty$ a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on $(\mathcal{E},F_\infty)$ equivalent to the "safely irrelevant conjecture". The Mordell-Weil group of $\mathcal{E}$ (with the Néron-Tate pairing) contains a canonical root system arising from $(-1)$-curves in special position in the Néron-Severi group. This canonical system is identified with the roots of the flavor group $\mathsf{F}$: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al.

Special Arithmetic of Flavor

TL;DR

The work reframes the rank-1 SW landscape through Diophantine geometry by treating special geometries as elliptic surfaces over the chiral ring, using the Kodaira-Néron model to classify non-free theories via the pair . It shows that SW completeness aligns with the safely irrelevant conjecture and that the flavor content is encoded in the Mordell-Weil lattice, with -root curves yielding flavor roots through good-position integral/narrow sections; the Oguiso-Shioda tables then determine admissible flavor groups. The analysis connects discrete gaugings to base changes of elliptic surfaces, and provides a comprehensive, geometry-driven account that agrees with Argyres et al. across known rank-1 classifications. The framework unifies UV behavior, flavor symmetry, and deformation data into a coherent arithmetic picture, enabling systematic exploration of discrete gauging and flavor-root structures via rational elliptic surfaces and their Mordell-Weil lattices.

Abstract

We revisit the classification of rank-1 4d QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-Néron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs where is a relatively minimal, rational elliptic surface with section, and a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on equivalent to the "safely irrelevant conjecture". The Mordell-Weil group of (with the Néron-Tate pairing) contains a canonical root system arising from -curves in special position in the Néron-Severi group. This canonical system is identified with the roots of the flavor group : the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al.

Paper Structure

This paper contains 43 sections, 7 theorems, 121 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. A sketch of flavor symmetry in rank-1 4d ${\cal N}=2$ QFT
  3. Organization of the paper.
  4. Special geometry and rational elliptic surfaces
  5. Preliminaries
  6. Rank-1 special geometries as rational elliptic surfaces
  7. Kodaira-Néron models.
  8. The chiral ring $\mathscr{R}$ is free.
  9. The functional invariant $\mathscr{J}$.
  10. Scale-invariant vs. mass-deformed special geometries.
  11. The surface ${\cal E}$ is rational.
  12. ${\cal E}$ and the symplectic structure of $(X,\Omega)$.
  13. Moduli of rational elliptic surfaces with given singular fibers.
  14. $ADE$ and all that.
  15. Allowed fiber configurations and Dynkin theorem.
  16. ...and 28 more sections

Key Result

Theorem 1

Let ${\cal E}$ be a (relatively minimal) rational elliptic surface. The following sequence (of finitely-generated Abelian groups) is exact In particular, the Shioda-Tate formula holds In addition, using rrtyu, the map $\gamma$ factors through $\mathsf{MW}({\cal E})$ so we get a map which is injective on the torsion subgroup.

Figures (2)

  • Figure 1: The $G_2$ Dynkin graph as a folding of the $D_4$ one.
  • Figure 2: Diagram folding $E_6\to F_4$.

Theorems & Definitions (52)

  • Remark 1
  • Definition 1
  • Remark 2
  • Definition 2
  • Definition 2
  • Remark 3
  • Definition 3
  • Remark 4
  • Example 1
  • Example 2
  • ...and 42 more