Abstract
We construct special Lagrangian 3-spheres in non-Kähler compact threefolds equipped with the Fu–Li–Yau geometry. 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.
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Notes
Here, we use the target space description of D-branes. In this description, also used e.g. in homological mirror symmetry, D-branes can be understood as objects of a certain category associated to X, e.g. the category of coherent sheaves on X. A rather different description of D-branes, from the world-sheet perspective, is based on boundary conditions in two-dimensional A-model or B-model (with X as a target manifold). The latter is used e.g. in the enumerative version of mirror symmetry that involves Gromov-Witten invariants on the A-model side. The relation between these two descriptions of D-branes is not at all obvious, especially at the mathematical level of rigor. In the present discussion, though, we only use the target space perspective.
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We thank M. Garcia-Fernandez for helpful comments. We also thank the referees for a careful reading.
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T.C.C is supported in part by NSF CAREER Grant DMS-194452 and an Alfred P. Sloan Fellowship.
S.G. is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, and by the National Science Foundation under Grant No. NSF DMS 1664227.
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Collins, T.C., Gukov, S., Picard, S. et al. Special Lagrangian Cycles and Calabi-Yau Transitions. Commun. Math. Phys. 401, 769–802 (2023). https://doi.org/10.1007/s00220-023-04655-3
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DOI: https://doi.org/10.1007/s00220-023-04655-3