The Weil correspondence and universal special geometry

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 $$ \mathcal{N} $$ 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 $$ \mathcal{B} $$ 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 $$ \mathcal{N} $$ N = 2 SCFT in terms of the Mordell-Weil lattice (with Néron-Tate height) of the Albanese variety $$ {A}_{\mathbbm{L}} $$ A L of the universal geometry seen as a quasi-Abelian variety $$ {Y}_{\mathbbm{L}} $$ Y L defined over the function field $$ \mathbbm{L}\equiv \mathbb{C}\left(\mathcal{B}\right) $$ L ≡ ℂ B .