Topological mirror symmetry with fluxes

TL;DR

This work constructs topological mirrors for Calabi–Yau manifolds with NS flux by exchanging flux quanta with torsion in the mirror’s cohomology, enriching the SYZ framework with a topological counterpart to differential mirror rules. It develops a concrete framework using Leray–Hirsch-type spectral sequences and a specialized basis of forms that captures the non-closure induced by flux, enabling KK reductions on SU(3) structure manifolds. A detailed example—the distorted mirror quintic—illustrates how NS flux yields torsion and how a mass spectrum can be organized into a topology-driven massless sector plus controlled massive modes, in alignment with Wall–Z̆ubr classification. The paper argues that in the infrared limit, the massless sector encodes topology while massive states carry no extra topological information, highlighting a deep link between topology and low-energy effective theories in SU(3) compactifications.

Abstract

Motivated by SU(3) structure compactifications, we show explicitly how to construct half--flat topological mirrors to Calabi--Yau manifolds with NS fluxes. Units of flux are exchanged with torsion factors in the cohomology of the mirror; this is the topological complement of previous differential--geometric mirror rules. The construction modifies explicit SYZ fibrations for compact Calabi--Yaus. The results are of independent interest for SU(3) compactifications. For example one can exhibit explicitly which massive forms should be used for Kaluza--Klein reduction, proving previous conjectures. Formality shows that these forms carry no topological information; this is also confirmed by infrared limits and old classification theorems.

Figures (5)

• Figure 1: Amoebas in one and two complex dimensions.
• Figure 2: The discrimimant locus $\Delta$ and the singular fibres over it. Also shown are the regular $T^3$ fibres over points outside $\Delta$.
• Figure 3: In green, a three--cycle which projects to a path (left), and its mirror (right).
• Figure 4: In green, a three--cycle which projects to an hexagon (left), and its mirror (right).
• Figure 5: The three--form $H$ is switched on along the red arrows in the quintic (left). The distorted mirror quintic is defined using a "rugby ball" neighborhood (right). The twisting is on two--chains in the base which intersects the path transversely. In dashed red is shown one such a transversal disc. Along the boundary of this neighborhood, the internal $T^3$s are glued to the external $T^3$s with a twisting of an $S^1\subset T^3$, according to the rule $\tilde{\phi}^1 \sim \phi^1 + N\theta$. The locus $\phi_1=$const is shown in fibres over the internal and external edges, at three different values of $\theta$ (down). The two--chain, in differently dashed red, continues away from the rugby ball.