Special Lagrangian cycles and Calabi-Yau transitions. (English) Zbl 1518.53065
The authors consider special submanifolds of Calabi-Yau threefolds. They consider holomorphic \(2\)-cycles and special Lagrangian \(3\)-cycles and construct special Lagrangian \(3\)-spheres in non-Kähler compact threefolds equipped with the Fu-Li-Yau geometry. This construction establishes the existence of smooth special Lagrangian cycles with respect to a balanced metric on any compact threefold emerging from a conifold transition. These non-Kähler geometries emerge from topological transitions of compact Calabi-Yau threefolds. From this point of view, a conifold transition exchanges holomorphic \(2\)-cycles for special Lagrangian \(3\)-cycles.
The problem discussed in this paper has direct applications to string theory, in the context of flux compactifications and compactifications on a \(6\)-manifold with \(\mathrm{SU}(3)\) structure. More specifically, the authors discuss the relation between \(\mathrm{SU}(3)\) structures and flux compactifications in physics.
The problem discussed in this paper has direct applications to string theory, in the context of flux compactifications and compactifications on a \(6\)-manifold with \(\mathrm{SU}(3)\) structure. More specifically, the authors discuss the relation between \(\mathrm{SU}(3)\) structures and flux compactifications in physics.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 53D42 | Symplectic field theory; contact homology |
| 14J42 | Holomorphic symplectic varieties, hyper-Kähler varieties |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 32Q25 | Calabi-Yau theory (complex-analytic aspects) |
| 57R17 | Symplectic and contact topology in high or arbitrary dimension |
| 81T33 | Dimensional compactification in quantum field theory |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
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