On the supergravity formulation of mirror symmetry in generalized Calabi-Yau manifolds

TL;DR

The paper addresses how to formulate the scalar potential for flux compactifications on generalized Calabi–Yau manifolds within $\mathcal{N}=2$ supergravity, incorporating both electric and magnetic gaugings of an abelian Heisenberg isometry. The authors build a duality-covariant framework by introducing an embedding tensor $Q$ that combines electric and magnetic charges, dualize certain axions to tensor fields, and integrate out massive degrees of freedom to obtain a manifestly mirror-symmetric effective potential $V_{eff}$ that preserves $Sp(2h_2+2)\times Sp(2h_1+2)$. The potential can be written in an $\mathcal{N}=1$ form with the Berglund–Mayr superpotential $W = e^{-(K_{SK}+K_{KS})/2}\, V_2^T Q \mathbb{C} V_1$, connecting the $\mathcal{N}=2$ gauged theory to the $\mathcal{N}=1$ landscapes and clarifying the duality structure of generalized geometries. Altogether, the work provides a concrete, symmetric effective description of flux-induced potentials in generalized Calabi–Yau compactifications and highlights a precise bridge to mirror-symmetric $\mathcal{N}=1$ formulations.

Abstract

We derive the complete supergravity description of the N=2 scalar potential which realizes a generic flux-compactification on a Calabi-Yau manifold (generalized geometry). The effective potential V_{eff}=V_{(\partial_Z V=0)}$, obtained by integrating out the massive axionic fields of the special quaternionic manifold, is manifestly mirror symmetric, i.e. invariant with respect to {\rm Sp}(2 h_2+2)\times {\rm Sp}(2 h_1+2) and their exchange, being h_1, h_2 the complex dimensions of the underlying special geometries. {\Scr V}_{eff} has a manifestly N=1 form in terms of a mirror symmetric superpotential $W$ proposed, some time ago, by Berglund and Mayr.