4-FLUXES AND NONPERTURBATIVE SUPERPOTENTIALS IN 2 DIMENSIONS

We show how certain non-perturbative superpotentials (W) over bar(Sigm a), which are the two-dimensional analogs of the Seiberg-Witten prepot ential 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 IP1 fibered over IP2. In the field theory l imit, this gives an effective U(1) gauge theory with N = (2,2) supersy mmetry 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 article is a shortened version of ref. [1].