Generalized Calabi-Yau manifolds
Nigel Hitchin
TL;DR
The paper unifies Calabi–Yau and symplectic geometries through generalized complex geometry, defining generalized complex and generalized Calabi–Yau structures via closed pure spinors and the Courant bracket with B-field automorphisms. In six dimensions it develops a variational approach with a volume functional whose critical points correspond to generalized Calabi–Yau structures, and proves a local moduli space exists as an open set in $H^{\mathrm{ev/od}}(M,\mathbf{R})$ under the $dd^J$-lemma or Lefschetz-type conditions. A natural special pseudo-Kähler structure is induced on the moduli space, and the framework accommodates twists by gerbes, yielding twisted generalized Calabi–Yau structures. The results illuminate how B-fields interpolate between symplectic and Calabi–Yau regimes and provide a rigorous foundation for background geometry in string-theoretic contexts, with concrete six-dimensional structure classifications. Overall, the work establishes a robust, cohomology-driven approach to global moduli, stability, and geometry of generalized Calabi–Yau structures.
Abstract
A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphisms and closed 2-forms. In the special case of six dimensions we characterize them as critical points of a natural variational problem on closed forms, and prove that a local moduli space is provided by an open set in either the odd or even cohomology.