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The SYZ Conjecture via Homological Mirror Symmetry

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Superschool on Derived Categories and D-branes (SDCD 2016)

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Abstract

These are expanded notes based on a talk given at the Superschool on Derived Categories and D-branes held at the University of Alberta in July of 2016. The goal of these notes is to give a motivated introduction to the Strominger-Yau-Zaslow (SYZ) conjecture from the point of view of homological mirror symmetry.

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Notes

  1. 1.

    One should work with the dg/\(A_\infty \) enhancements of these categories but we ignore that here.

  2. 2.

    For background on D-branes see for example [2] or the other entries in this volume.

  3. 3.

    D0, D3, ...denote 0-dimensional, 3-dimensional, ...D-branes.

  4. 4.

    That is, k in degree zero and 0 in other degrees.

  5. 5.

    This choice is the reason that X may have several mirrors.

  6. 6.

    That is, a degeneration with maximally unipotent monodromy. These are sometimes known as large complex structure limits (LCSL).

  7. 7.

    Here we’ve assumed for simplicity that the only critical value of W is at \(0 \in {\mathbb C}\).

  8. 8.

    More precisely, the sum is over curve classes \(\beta \) with Maslov index \(\mu (\beta ) = 2\).

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Authors and Affiliations

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

    Dori Bejleri

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  1. Dori Bejleri
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Correspondence to Dori Bejleri .

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Editors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Matthew Ballard

  2. Department of Mathematics and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada

    Charles Doran

  3. Department of Mathematics and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada

    David Favero

  4. Department of Physics, Virginia Tech, Blacksburg, Virginia, USA

    Eric Sharpe

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Bejleri, D. (2018). The SYZ Conjecture via Homological Mirror Symmetry. In: Ballard, M., Doran, C., Favero, D., Sharpe, E. (eds) Superschool on Derived Categories and D-branes. SDCD 2016. Springer Proceedings in Mathematics & Statistics, vol 240. Springer, Cham. https://doi.org/10.1007/978-3-319-91626-2_13

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