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
HTML articles powered by AMS MathViewer

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

PDF | Request permission

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

Mohammed Abouzaid, Morse homology, tropical geometry, and homological mirror symmetry for toric varieties, Selecta Math. (N.S.) 15 (2009), no. 2, 189–270. MR 2529936, DOI 10.1007/s00029-009-0492-2
Mohammed Abouzaid, A geometric criterion for generating the Fukaya category, Publ. Math. Inst. Hautes Études Sci. 112 (2010), 191–240. MR 2737980, DOI 10.1007/s10240-010-0028-5
Mohammed Abouzaid, A cotangent fibre generates the Fukaya category, Adv. Math. 228 (2011), no. 2, 894–939. MR 2822213, DOI 10.1016/j.aim.2011.06.007
M. Abouzaid, D. Auroux, L. Katzarkov, Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces, arXiv:1205.0053.
M. Abouzaid, K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, in preparation.
Mohammed Abouzaid and Paul Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010), no. 2, 627–718. MR 2602848, DOI 10.2140/gt.2010.14.627
Mohammed Abouzaid and Ivan Smith, Homological mirror symmetry for the 4-torus, Duke Math. J. 152 (2010), no. 3, 373–440. MR 2654219, DOI 10.1215/00127094-2010-015
Denis Auroux, Ludmil Katzarkov, and Dmitri Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, Ann. of Math. (2) 167 (2008), no. 3, 867–943. MR 2415388, DOI 10.4007/annals.2008.167.867
Denis Auroux, Ludmil Katzarkov, and Dmitri Orlov, Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves, Invent. Math. 166 (2006), no. 3, 537–582. MR 2257391, DOI 10.1007/s00222-006-0003-4
R. Bocklandt, Noncommutative mirror symmetry for punctured surfaces, arXiv:1111.3392.
A. Bondal, W. D. Ruan, Mirror symmetry for weighted projective spaces, in preparation.
Alexander I. Efimov, Homological mirror symmetry for curves of higher genus, Adv. Math. 230 (2012), no. 2, 493–530. MR 2914956, DOI 10.1016/j.aim.2012.02.022
Kwokwai Chan and Naichung Conan Leung, Mirror symmetry for toric Fano manifolds via SYZ transformations, Adv. Math. 223 (2010), no. 3, 797–839. MR 2565550, DOI 10.1016/j.aim.2009.09.009
Cheol-Hyun Cho, Products of Floer cohomology of torus fibers in toric Fano manifolds, Comm. Math. Phys. 260 (2005), no. 3, 613–640. MR 2183959, DOI 10.1007/s00220-005-1421-7
Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, and Eric Zaslow, T-duality and homological mirror symmetry for toric varieties, Adv. Math. 229 (2012), no. 3, 1875–1911. MR 2871160, DOI 10.1016/j.aim.2011.10.022
Kenji Fukaya, Mirror symmetry of abelian varieties and multi-theta functions, J. Algebraic Geom. 11 (2002), no. 3, 393–512. MR 1894935, DOI 10.1090/S1056-3911-02-00329-6
K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian Floer theory on compact toric manifolds: survey, Surveys in Differential Geometry, vol. 17, “Algebra and Geometry: In Memory of C. C. Hsiung”, International Press, 2012, pp. 229–298.
K. Fukaya, P. Seidel, and I. Smith, The symplectic geometry of cotangent bundles from a categorical viewpoint, Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 1–26. MR 2596633, DOI 10.1007/978-3-540-68030-7_{1}
S. Ganatra, Symplectic cohomology and duality for the wrapped Fukaya category, Ph.D. thesis, MIT, 2012.
M. Gross, L. Katzarkov, H. Ruddat, Towards mirror symmetry for varieties of general type, arXiv: 1202.4042.
K. Hori, C. Vafa, Mirror symmetry, arXiv:hep-th/0002222.
Anton Kapustin, Ludmil Katzarkov, Dmitri Orlov, and Mirroslav Yotov, Homological mirror symmetry for manifolds of general type, Cent. Eur. J. Math. 7 (2009), no. 4, 571–605. MR 2563433, DOI 10.2478/s11533-009-0056-x
Anton Kapustin and Dmitri Orlov, Vertex algebras, mirror symmetry, and D-branes: the case of complex tori, Comm. Math. Phys. 233 (2003), no. 1, 79–136. MR 1957733, DOI 10.1007/s00220-002-0755-7
Bernhard Keller, Introduction to $A$-infinity algebras and modules, Homology Homotopy Appl. 3 (2001), no. 1, 1–35. MR 1854636, DOI 10.4310/hha.2001.v3.n1.a1
Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151–190. MR 2275593
Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918
M. Kontsevich, Lectures at ENS, Paris, Spring 1998, notes taken by J. Bellaiche, J.-F. Dat, I. Marin, G. Racinet, and H. Randriambololona, unpublished.
Maxim Kontsevich and Yan Soibelman, Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 203–263. MR 1882331, DOI 10.1142/9789812799821_{0}007
M. Kontsevich and Y. Soibelman, Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry, Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 153–219. MR 2596638
K. H. Lin, D. Pomerleano, Global matrix factorizations, arXiv:1101.5847.
Valery A. Lunts, Categorical resolution of singularities, J. Algebra 323 (2010), no. 10, 2977–3003. MR 2609187, DOI 10.1016/j.jalgebra.2009.12.023
D. O. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 240–262 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3(246) (2004), 227–248. MR 2101296
Dmitri Orlov, Matrix factorizations for nonaffine LG-models, Math. Ann. 353 (2012), no. 1, 95–108. MR 2910782, DOI 10.1007/s00208-011-0676-x
Alexander Polishchuk and Eric Zaslow, Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys. 2 (1998), no. 2, 443–470. MR 1633036, DOI 10.4310/ATMP.1998.v2.n2.a9
A. Preygel, Thom-Sebastiani Duality for Matrix Factorizations, arXiv:1101.5834.
Marco Schlichting, Negative $K$-theory of derived categories, Math. Z. 253 (2006), no. 1, 97–134. MR 2206639, DOI 10.1007/s00209-005-0889-3
P. Seidel, Homological mirror symmetry for the quartic surface, arXiv:math/0310414.
Paul Seidel, Fukaya categories and deformations, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 351–360. MR 1957046
Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780, DOI 10.4171/063
Paul Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geom. 20 (2011), no. 4, 727–769. MR 2819674, DOI 10.1090/S1056-3911-10-00550-3
P. Seidel, Some speculations on Fukaya categories and pair-of-pants decompositions, Surveys in Differential Geometry, vol. 17, “Algebra and Geometry: In Memory of C. C. Hsiung”, International Press, 2012, pp. 411–426.
Nick Sheridan, On the homological mirror symmetry conjecture for pairs of pants, J. Differential Geom. 89 (2011), no. 2, 271–367. MR 2863919
N. Sheridan, Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space, arXiv: 1111.0632.
Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243–259. MR 1429831, DOI 10.1016/0550-3213(96)00434-8
R. W. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), no. 1, 1–27. MR 1436741, DOI 10.1023/A:1017932514274
Kazushi Ueda, Homological mirror symmetry for toric del Pezzo surfaces, Comm. Math. Phys. 264 (2006), no. 1, 71–85. MR 2212216, DOI 10.1007/s00220-005-1509-0
Charles Weibel, The negative $K$-theory of normal surfaces, Duke Math. J. 108 (2001), no. 1, 1–35. MR 1831819, DOI 10.1215/S0012-7094-01-10811-9