Generalized Calabi-Yau structures and mirror symmetry
Claus Jeschek
TL;DR
Problem addressed: traditional mirror symmetry in fluxless Calabi–Yau backgrounds does not address how B-fields and algebraic structures map under duality. Approach: recast backgrounds in Hitchin–Gualtieri generalized complex geometry and define a mirror map ${\mathcal M}$ together with $B$-field transformations $e^B$, then test on the $T^6$ example. Key results: the mirror map exchanges not only the metric and $B$-field but also the generalized complex data (i.e., pure spinors) and reproduces the Buscher rules without ad hoc input; explicit mirror data are derived for both $B=0$ and $B\neq0$ cases. Generalization: the framework extends to more general fibrations and potential NS-flux via twisted differential $d^H$, with D-branes interpreted as generalized submanifolds. Significance: provides a unified, mathematically natural description of mirror symmetry with fluxes and torsion, with potential applications in flux compactifications and beyond.
Abstract
We use the differential geometrical framework of generalized (almost) Calabi-Yau structures to reconsider the concept of mirror symmetry. It is shown that not only the metric and B-field but also the algebraic structures are uniquely mapped. As an example we use the six-torus as a trivial generalized Calabi-Yau 6-fold and an appropriate B-field.