The Weil Correspondence and Universal Special Geometry

TL;DR

This paper explores the geometric and algebraic structures underlying Seiberg-Witten theory, connecting the Weil correspondence, universal special geometry, and flavor symmetries through algebraic integrable systems and abelian varieties.

Contribution

It introduces the universal special geometry as an algebraic integrable system with anti-affine extensions, providing a new geometric framework for understanding Seiberg-Witten and flavor symmetries.

Findings

Universal special geometry characterized by anti-affine extensions.

Seiberg-Witten geometries as symplectic quotients of the universal system.

Flavor symmetry analyzed via Mordell-Weil lattice of the Albanese variety.

Abstract

The Weil correspondence states that the datum of a Seiberg-Witten differential is equivalent to an algebraic group extension of the integrable system associated to the Seiberg-Witten geometry. Remarkably this group extension represents quantum consistent couplings for the $\mathcal{N}=2$ QFT if and only if the extension is anti-affine in the algebro-geometric sense. The universal special geometry is the algebraic integrable system whose Lagrangian fibers are the anti-affine extension groups; it is defined over a base $\mathscr{B}$ parametrized by the Coulomb coordinates and the couplings. On the total space of the universal geometry there is a canonical (holomorphic) Euler differential. The ordinary Seiberg-Witten geometries at fixed couplings are symplectic quotients of the universal one, and the Seiberg-Witten differential arises as the reduction of the Euler one in accordance with the Weil correspondence. This universal viewpoint allows to study geometrically the flavor symmetry of the $\mathcal{N}=2$ SCFT in terms of the Mordell-Weil lattice (with N\'eron-Tate height) of the Albanese variety $A_\mathbb{L}$ of the universal geometry seen as a quasi-Abelian variety $Y_\mathbb{L}$ defined over the function field $\mathbb{L}\equiv\mathbb{C}(\mathscr{B})$.