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A Lie theoretical construction of a Landau–Ginzburg model without projective mirrors

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Abstract

We describe the Fukaya–Seidel category of a Landau–Ginzburg model \(\mathrm {LG}(2)\) for the semisimple adjoint orbit of \(\mathfrak {sl}(2, {\mathbb {C}})\). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to \(\mathrm {LG}(2)\), and that this remains so after compactification.

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Acknowledgements

We are grateful to Patrick Clarke for pointing out a significant improvement to an earlier version of this work. We thank Denis Auroux, Lutz Hille, Ludmil Katzarkov, and Sukhendu Mehrotra for helpful suggestions and comments. S. Barmeier is supported by the Studienstiftung des deutschen Volkes. Part of this work was completed during a visit of L. Grama to Chile. We are thankful to the Vice Rectoría de Investigación and Desarrollo Tecnológico of the Universidad Católica del Norte whose support made this visit possible. L. Grama is partially supported by FAPESP Grant 2016/22755-1. E. Gasparim was partially supported by a Simons Associateship ICTP, and Network Grant NT8, Office of External Activities, ICTP, Italy. E. Ballico was partially supported by MIUR and GNSAGA of INdAM (Italy).

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Authors and Affiliations

  1. Department of Mathematics, University of Trento, 38050, Povo, Trento, Italy

    E. Ballico

  2. Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149, Münster, Germany

    S. Barmeier

  3. Departamento de Matemáticas, Universidad Católica del Norte, Av. Angamos 0600, Antofagasta, Chile

    E. Gasparim

  4. Imecc – Unicamp, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz, Campinas, SP, 13083–859, Brasil

    L. Grama & L. A. B. San Martin

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  1. E. Ballico
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  2. S. Barmeier
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  3. E. Gasparim
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  4. L. Grama
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  5. L. A. B. San Martin
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Correspondence to E. Gasparim.

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Ballico, E., Barmeier, S., Gasparim, E. et al. A Lie theoretical construction of a Landau–Ginzburg model without projective mirrors. manuscripta math. 158, 85–101 (2019). https://doi.org/10.1007/s00229-018-1024-1

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  • DOI: https://doi.org/10.1007/s00229-018-1024-1

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