Fayet-Iliopoulos Potentials from Four-Folds

TL;DR

This paper explores how non-perturbative superpotentials in 2D theories can be derived from geometric engineering using 4-folds, revealing connections between FI potentials, modular parameters, and supersymmetry breaking.

Contribution

It demonstrates the computation of superpotentials from 4-fold geometries and links FI couplings to the modular parameter of a K3 surface, providing new insights into supersymmetry breaking mechanisms.

Findings

FI potential relates to the middle period of a meromorphic differential.

Supersymmetry is non-perturbatively broken when 4-flux is present.

Tuning the FI coupling can restore supersymmetry at a singular K3.

Abstract

We show how certain non-perturbative superpotentials W, which are the two-dimensional analogs of the Seiberg-Witten prepotential in 4d, can be computed via geometric engineering from 4-folds. We analyze an explicit example for which the relevant compact geometry of the 4-fold is given by $P^1$ fibered over $P^2$. In the field theory limit, this gives an effective U(1) gauge theory with N=(2,2) supersymmetry in two dimensions. We find that the analog of the SW curve is a K3 surface, and that the complex FI coupling is given by the modular parameter of this surface. The FI potential itself coincides with the middle period of a meromorphic differential. However, it only shows up in the effective action if a certain 4-flux is switched on, and then supersymmetry appears to be non-perturbatively broken. This can be avoided by tuning the bare FI coupling by hand, in which case the supersymmetric minimum naturally corresponds to a singular K3.