On the twisted chiral potential in 2d and the analogue of rigid special geometry for 4-folds

TL;DR

This paper explores the non-perturbative twisted chiral potential in 2D supersymmetric gauge theories derived from Calabi-Yau 4-folds, extending concepts of special geometry to complex four-dimensional geometries.

Contribution

It introduces a method to compute the twisted chiral potential from Calabi-Yau 4-folds and tests the analogue of rigid special geometry in this context.

Findings

Explicit computation of the twisted chiral potential from middle periods.

Verification of the analogue of rigid special geometry for Calabi-Yau 4-folds.

Identification of the geometric origin of the potential in the 4-fold structure.

Abstract

We discuss how to obtain an N=(2,2) supersymmetric SU(3) gauge theory in two dimensions via geometric engineering from a Calabi-Yau 4-fold and compute its non-perturbative twisted chiral potential. The relevant compact part of the 4-fold geometry consists of two intersecting P^1's fibered over P^2. The rigid limit of the local mirror of this geometry is a complex surface that generalizes the Seiberg-Witten curve and on which there exist two holomorphic 2-forms. These stem from the same meromorphic 2-form as derivatives w.r.t. the two moduli, respectively. The middle periods of this meromorphic form give directly the twisted chiral potential. The explicit computation of these and of the four-point Yukawa couplings allows for a non-trivial test of the analogue of rigid special geometry for a 4-fold with several moduli.