Geometric Engineering of N=1 Quantum Field Theories

TL;DR

This paper develops a geometric framework using F- and M-theory compactifications on Calabi-Yau fourfolds to model N=1 supersymmetric gauge theories, analyzing superpotentials generated by instantons and their relation to Toda theories.

Contribution

It introduces a novel geometric construction for N=1 gauge theories and computes the superpotential as a function of compactification radius, linking it to affine Toda theories.

Findings

Superpotential matches affine Toda potential at finite radius

Superpotential reduces to Toda potential as radius approaches zero

Provides a geometric realization of N=1 gauge theories in string theory

Abstract

We construct local geometric model in terms of F- and M-theory compactification on Calabi-Yau fourfolds which lead to N=1 Yang-Mills theory in d=4 and its reduction on a circle to d=3. We compute the superpotential in d=3, as a function of radius, which is generated by the Euclidean 5-brane instantons. The superpotential turns out to be the same as the potential for affine Toda theories. In the limit of vanishing radius the affine Toda potential reduces to the Toda potential.