## 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.## References

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## Bibliographic Information

**Mohammed Abouzaid**- Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
- MR Author ID: 734175
- Email: abouzaid@math.columbia.edu
**Denis Auroux**- Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720-3840
- Email: auroux@math.berkeley.edu
**Alexander I. Efimov**- Affiliation: Algebraic Geometry Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin Street, Moscow 119991, Russia
- Email: efimov13@yandex.ru
**Ludmil Katzarkov**- Affiliation: Department of Mathematics, Universität Wien, Garnisongasse 3, Vienna A-1090, Austria, and University of Miami, P.O. Box 249085, Coral Gables, Florida 33124-4250
- MR Author ID: 346264
- Email: lkatzark@math.uci.edu
**Dmitri Orlov**- Affiliation: Algebraic Geometry Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin Street, Moscow 119991, Russia
- Email: orlov@mi.ras.ru
- Received by editor(s): March 22, 2011
- Received by editor(s) in revised form: March 2, 2013
- Published electronically: April 4, 2013
- Additional Notes: The first author was supported by a Clay Research Fellowship

The second author was partially supported by NSF grants DMS-0652630 and DMS-1007177

The third author was partially supported by the Dynasty Foundation, NSh grant 4713.2010.1, and by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023

The fourth author was funded by NSF grant DMS-0600800, NSF FRG grant DMS-0652633, FWF grant P20778, and an ERC grant – GEMIS

The last author was partially supported by RFBR grants 10-01-93113, 11-01-00336, 11-01-00568, NSh grant 4713.2010.1, and by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023 - © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**26**(2013), 1051-1083 - MSC (2010): Primary 53D37, 14J33; Secondary 53D40, 53D12, 18E30, 14F05
- DOI: https://doi.org/10.1090/S0894-0347-2013-00770-5
- MathSciNet review: 3073884