Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Noncommutative mirror symmetry for punctured surfaces


Author: Raf Bocklandt; With an appendix by Mohammed Abouzaid
Journal: Trans. Amer. Math. Soc. 368 (2016), 429-469
MSC (2010): Primary 16G20, 14J33
DOI: https://doi.org/10.1090/tran/6375
Published electronically: April 3, 2015
MathSciNet review: 3413869
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 2013, Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities. We generalize this result to other punctured Riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models. In particular, given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is $ \mathtt {A}_\infty $-isomorphic to a subcategory of the wrapped Fukaya category of a punctured Riemann surface. The connection between the dimer model and the punctured Riemann surface then has a nice interpretation in terms of a duality on dimer models.


References [Enhancements On Off] (What's this?)

  • [1] 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 (2011h:53123), https://doi.org/10.1007/s00029-009-0492-2
  • [2] Mohammed Abouzaid, Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, and Dmitri Orlov, Homological mirror symmetry for punctured spheres, J. Amer. Math. Soc. 26 (2013), no. 4, 1051-1083. MR 3073884, https://doi.org/10.1090/S0894-0347-2013-00770-5
  • [3] Mohammed Abouzaid and Paul Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010), no. 2, 627-718. MR 2602848 (2011g:53190), https://doi.org/10.2140/gt.2010.14.627
  • [4] Ibrahim Assem, Thomas Brüstle, Gabrielle Charbonneau-Jodoin, and Pierre-Guy Plamondon, Gentle algebras arising from surface triangulations, Algebra Number Theory 4 (2010), no. 2, 201-229. MR 2592019 (2011h:16045), https://doi.org/10.2140/ant.2010.4.201
  • [5] Ibrahim Assem and Andrzej Skowroński, Iterated tilted algebras of type $ \tilde {\bf A}_n$, Math. Z. 195 (1987), no. 2, 269-290. MR 892057 (88m:16033), https://doi.org/10.1007/BF01166463
  • [6] 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 (2009f:53142), https://doi.org/10.4007/annals.2008.167.867
  • [7] 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 (2007g:14045), https://doi.org/10.1007/s00222-006-0003-4
  • [8] Paul Balmer and Marco Schlichting, Idempotent completion of triangulated categories, J. Algebra 236 (2001), no. 2, 819-834. MR 1813503 (2002a:18013), https://doi.org/10.1006/jabr.2000.8529
  • [9] Michael J. Bardzell, The alternating syzygy behavior of monomial algebras, J. Algebra 188 (1997), no. 1, 69-89. MR 1432347 (98a:16009), https://doi.org/10.1006/jabr.1996.6813
  • [10] Raf Bocklandt, Calabi-Yau algebras and weighted quiver polyhedra, Math. Z. 273 (2013), no. 1-2, 311-329. MR 3010162, https://doi.org/10.1007/s00209-012-1006-z
  • [11] Raf Bocklandt, Consistency conditions for dimer models, Glasg. Math. J. 54 (2012), no. 2, 429-447. MR 2911380, https://doi.org/10.1017/S0017089512000080
  • [12] Nathan Broomhead, Dimer models and Calabi-Yau algebras, Mem. Amer. Math. Soc. 215 (2012), no. 1011, viii+86. MR 2908565, https://doi.org/10.1090/S0065-9266-2011-00617-9
  • [13] Ben Davison, Consistency conditions for brane tilings, J. Algebra 338 (2011), 1-23. MR 2805177 (2012e:14110), https://doi.org/10.1016/j.jalgebra.2011.05.005
  • [14] Bo Feng, Yang-Hui He, Kristian D. Kennaway, and Cumrun Vafa, Dimer models from mirror symmetry and quivering amoebae, Adv. Theor. Math. Phys. 12 (2008), no. 3, 489-545. MR 2399318 (2009k:81180)
  • [15] Sebastián Franco, Amihay Hanany, David Vegh, Brian Wecht, and Kristian D. Kennaway, Brane dimers and quiver gauge theories, J. High Energy Phys. 1 (2006), 096, 48 pp. (electronic). MR 2201227 (2007b:81191), https://doi.org/10.1088/1126-6708/2006/01/096
  • [16] V. Ginzburg, Calabi-Yau algebras, math/0612139.
  • [17] Amihay Hanany, Christopher P. Herzog, and David Vegh, Brane tilings and exceptional collections, J. High Energy Phys. 7 (2006), 001, 44 pp. (electronic). MR 2240899 (2008b:81224), https://doi.org/10.1088/1126-6708/2006/07/001
  • [18] A. Hanany and K. D. Kennaway, Dimer models and toric diagrams, hep-th/0602041.
  • [19] Amihay Hanany and David Vegh, Quivers, tilings, branes and rhombi, J. High Energy Phys. 10 (2007), 029, 35. MR 2357949 (2008m:81153), https://doi.org/10.1088/1126-6708/2007/10/029
  • [20] Allen Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991), no. 2, 189-194. MR 1123262 (92f:57020), https://doi.org/10.1016/0166-8641(91)90050-V
  • [21] K. Hori, C. Vafa, Mirror symmetry, arXiv:hep-th/0002222.
  • [22] Daniel R. Gulotta, Properly ordered dimers, $ R$-charges, and an efficient inverse algorithm, J. High Energy Phys. 10 (2008), 014, 31. MR 2453031 (2010b:81116), https://doi.org/10.1088/1126-6708/2008/10/014
  • [23] Akira Ishii and Kazushi Ueda, On moduli spaces of quiver representations associated with dimer models, Higher dimensional algebraic varieties and vector bundles, RIMS Kôkyûroku Bessatsu, B9, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, pp. 127-141. MR 2509696 (2011e:16024)
  • [24] A. Ishii, K. Ueda, Dimer models and exceptional collections, arXiv:0911.4529
  • [25] Akira Ishii and Kazushi Ueda, A note on consistency conditions on dimer models, Higher dimensional algebraic geometry, RIMS Kôkyûroku Bessatsu, B24, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 143-164. MR 2809653
  • [26] T. V. Kadeishvili, The algebraic structure in the homology of an $ A(\infty )$-algebra, Soobshch. Akad. Nauk Gruzin. SSR 108 (1982), no. 2, 249-252 (1983) (Russian, with English and Georgian summaries). MR 720689 (84k:55009)
  • [27] Ludmil Katzarkov, Birational geometry and homological mirror symmetry, Real and complex singularities, World Sci. Publ., Hackensack, NJ, 2007, pp. 176-206. MR 2336686 (2008g:14062), https://doi.org/10.1142/9789812706898_0008
  • [28] Bernhard Keller, Introduction to $ A$-infinity algebras and modules, Homology Homotopy Appl. 3 (2001), no. 1, 1-35. MR 1854636 (2004a:18008a)
  • [29] Kristian D. Kennaway, Brane tilings, Internat. J. Modern Phys. A 22 (2007), no. 18, 2977-3038. MR 2343711 (2009a:81155), https://doi.org/10.1142/S0217751X07036877
  • [30] Richard Kenyon, An introduction to the dimer model, School and Conference on Probability Theory, ICTP Lect. Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 267-304 (electronic). MR 2198850 (2006k:82033)
  • [31] 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 (97f:32040)
  • [32] Maxim Kontsevich and Yan Soibelman, Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., vol. 21, Kluwer Acad. Publ., Dordrecht, 2000, pp. 255-307. MR 1805894 (2002e:18012)
  • [33] 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 (2011f:53183)
  • [34] S. Mozgovoy, Crepant resolutions and brane tilings I: Toric realization, arXiv:0908.3475
  • [35] M. Bender and S. Mozgovoy, Crepant resolutions and brane tilings II: Tilting bundles, arXiv:0909.2013
  • [36] Sergey Mozgovoy and Markus Reineke, On the noncommutative Donaldson-Thomas invariants arising from brane tilings, Adv. Math. 223 (2010), no. 5, 1521-1544. MR 2592501 (2011e:16026), https://doi.org/10.1016/j.aim.2009.10.001
  • [37] 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 (2006i:81173)
  • [38] D. O. Orlov, Triangulated categories of singularities, and equivalences between Landau-Ginzburg models, Mat. Sb. 197 (2006), no. 12, 117-132 (Russian, with Russian summary); English transl., Sb. Math. 197 (2006), no. 11-12, 1827-1840. MR 2437083 (2009g:14013), https://doi.org/10.1070/SM2006v197n12ABEH003824
  • [39] Alexander Quintero Vélez, McKay correspondence for Landau-Ginzburg models, Commun. Number Theory Phys. 3 (2009), no. 1, 173-208. MR 2504756 (2010b:14023), https://doi.org/10.4310/CNTP.2009.v3.n1.a4
  • [40] Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780 (2009f:53143)
  • [41] Michel van den Bergh, Non-commutative crepant resolutions, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749-770. MR 2077594 (2005e:14002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 16G20, 14J33

Retrieve articles in all journals with MSC (2010): 16G20, 14J33


Additional Information

Raf Bocklandt
Affiliation: Korteweg de Vries institute, University of Amsterdam (UvA), Science Park 904, 1098 XH Amsterdam, The Netherlands
Email: raf.bocklandt@gmail.com

Mohammed Abouzaid
Affiliation: Korteweg de Vries institute, University of Amsterdam (UvA), Science Park 904, 1098 XH Amsterdam, The Netherlands

DOI: https://doi.org/10.1090/tran/6375
Received by editor(s): December 20, 2011
Received by editor(s) in revised form: February 4, 2013, and November 12, 2013
Published electronically: April 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society