Type IIA and Heterotic String Vacua in D=2

TL;DR

The paper develops a detailed map between Type IIA string theory on Calabi–Yau fourfolds and heterotic string theory on Calabi–Yau threefolds times a torus in two dimensions, establishing a robust effective-theory duality for $D=2$, $N=(2,2)$ vacua. It shows that turning on RR flux in IIA generates a two-part superpotential $W$ and $\tilde{W}$, with $W$ tied to complex-structure data and $\tilde{W}$ to Kähler data; in the large-base limit for threefold-fibred fourfolds, $W(\tilde{Y}_4)$ reduces to a symplectic form $\alpha_I X^I - \beta^I F_I$ that matches the Taylor–Vafa IIB result, bridging to known BPS and black-hole structures. The analysis extends to $K3$-fibered geometries, where quantum corrections and modular symmetries are encoded via fiber/base prepotentials and homogeneous coordinates, and shows the heterotic dual captures these effects consistently. Together, these results illuminate how flux-induced potentials in lower-dimensional vacua are intertwined across dual descriptions and with established structures such as prepotentials and attractor physics, offering a coherent picture of non-perturbative dynamics in $D=2$ string vacua.

Abstract

We study type IIA string theory compactified on Calabi-Yau fourfolds and heterotic string theory compactified on Calabi-Yau threefolds times a two-torus. We derive the resulting effective theories which have two space-time dimensions and preserve four supercharges. The duality between such vacua is established at the level of the effective theory. For type IIA vacua with non-trivial Ramond-Ramond background fluxes a superpotential is generated. We show that for a specific choice of background fluxes and a fourfold which has the structure of a threefold fibred over a sphere the superpotential coincides with the superpotential recently proposed by Taylor and Vafa in compactifications of type IIB string theory on a threefold.