Mirror Symmetry in Generalized Calabi-Yau Compactifications

TL;DR

The paper extends mirror symmetry to generalized Calabi–Yau compactifications with NS flux, showing that Type IIB on a Calabi–Yau with NS flux $H_3$ is mirror to Type IIA on a half-flat $SU(3)$-structure manifold $\hat{Y}$. The NS flux is encoded geometrically via intrinsic torsion in the half-flat geometry, enabling a purely geometric mirror description and a consistent 4D $N=2$ effective action that matches the IIB NS-flux action. This is demonstrated through torus and SYZ analyses, domain-wall perspectives, and a detailed IIA reduction on $\hat{Y}$, including a nontrivial Ricci-scalar contribution to the potential. Overall, the work provides a concrete geometric realization of flux mirror symmetry and suggests a broad framework for dualities involving fluxes and generalized geometries, with implications for domains walls and $G_2$-holonomy constructions.

Abstract

We discuss mirror symmetry in generalized Calabi-Yau compactifications of type II string theories with background NS fluxes. Starting from type IIB compactified on Calabi-Yau threefolds with NS three-form flux we show that the mirror type IIA theory arises from a purely geometrical compactification on a different class of six-manifolds. These mirror manifolds have SU(3) structure and are termed half-flat; they are neither complex nor Ricci-flat and their holonomy group is no longer SU(3). We show that type IIA appropriately compactified on such manifolds gives the correct mirror-symmetric low-energy effective action.