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Special Lagrangian Cycles and Calabi-Yau Transitions

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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

  1. 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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Acknowledgements

We thank M. Garcia-Fernandez for helpful comments. We also thank the referees for a careful reading.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Tristan C. Collins

  2. Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, 91125, USA

    Sergei Gukov

  3. Department of Mathematics, UBC, 1984 Mathematics Road, Vancouver, BC, Canada

    Sebastien Picard

  4. Department of Mathematics, Tsinghua University, Haidian District, Beijing, 100084, China

    Shing-Tung Yau

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  1. Tristan C. Collins
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  4. Shing-Tung Yau
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Correspondence to Sebastien Picard.

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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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