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Convergence of Hermitian–Yang–Mills connections on two-dimensional Kähler tori and mirror symmetry

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We study the behavior of a family of Hermitian–Yang–Mills connections on complex two-dimensional Kähler tori, when the metrics degenerate. We prove a convergence theorem of connections for this family. From this result and an idea coming from mirror symmetry, we construct a (special) Lagrangian submanifold on the mirror torus. Conjecturally this gives the inverse of the homological mirror symmetry construction of Fukaya (J Algebraic Geom 11(3):393–512, 2002).

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Acknowledgements

I would like to thank Professor Kenji Fukaya for giving me invaluable suggestions and advice. I thank Professor Manabu Akaho for informing me about the erratum of [5] and the referee for careful reading and useful comments. The author is supported by JSPS KAKENHI Grant Number 26400061 and 18K03313.

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  1. Department of Mathematics, Rikkyo University, Nishi-Ikebukuro, Toshima, Tokyo, Japan

    Takeo Nishinou

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Correspondence to Takeo Nishinou.

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Nishinou, T. Convergence of Hermitian–Yang–Mills connections on two-dimensional Kähler tori and mirror symmetry. Lett Math Phys 111, 57 (2021). https://doi.org/10.1007/s11005-021-01405-1

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