Homological mirror symmetry for punctured spheres

Abouzaid, Mohammed; Auroux, Denis; Efimov, Alexander; Katzarkov, Ludmil; Orlov, Dmitri

doi:10.1090/S0894-0347-2013-00770-5

Homological mirror symmetry for punctured spheres
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by Mohammed Abouzaid, Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov and Dmitri Orlov

J. Amer. Math. Soc. 26 (2013), 1051-1083

DOI: https://doi.org/10.1090/S0894-0347-2013-00770-5

Published electronically: April 4, 2013

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Abstract:

We prove that the wrapped Fukaya category of a punctured sphere ($S^{2}$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.

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