Pure Gauge SU(2) Seiberg-Witten Theory and Modular Forms

TL;DR

This paper connects pure gauge SU(2) Seiberg-Witten theory with modular forms by identifying its spectral curve with a Weierstrass curve, enabling explicit expressions for key parameters and exploring links to conformal field theory.

Contribution

It explicitly identifies the spectral curve of SU(2) Seiberg-Witten theory with a Weierstrass curve and derives modular forms for key physical parameters, linking gauge theory and conformal field theory.

Findings

Explicit expression of the spectral curve as a Weierstrass curve.

Derivation of moduli space parameters from modular and theta functions.

Discussion of the relationship with the c=-2 triplet model CFT.

Abstract

We identify the spectral curve of pure gauge SU(2) Seiberg-Witten theory with the Weierstrass curve $\mathbbm{C}/L \ni z \mapsto (1,\wp(z),\wp(z)')$ and thereby obtain explicitely a modular form from which the moduli space parameter $u$ and lattice parameters $a$, $a_D$ can be derived in terms of modular respectively theta functions. We further discuss its relationship with the $c=-2$ triplet model conformal field theory.