Quantum tori, mirror symmetry and deformation theory

Letters in Mathematical Physics

Volumn 56, Issue 2, 2001, Pages 99-125

  Soibelman, Yan ^a  

^a KANSAS STATE UNIVERSITY   (United States)

Author keywords

A category; Abelian variety; Morita equivalence; Quantum torus

Indexed keywords

EID: 0038182402     PISSN: 03779017     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1010984821575     Document Type: Article

References (55)

1. Arinkin, D. and Polishchuk, A.: Fukaya category and Fourier transform, math.AG/9811023.
2. Barannikov, S.: Semi-infinite Hodge structures and mirror symmetry for projective spaces, math.AG/0010157.
3. Barannikov, S.: Quantum periods - I. Semi-infinite variations of Hodge structures, math.AG/0006193.
4. Barannikov, S.: Generalized periods and mirror symmetry in dimensions n > 3, math.AG/9903124.
5. Barannikov, S. and Kontsevich, M.: Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices (1998), No. 4, 201-215 (alg-geom/9710032).
6. Baranovsky, V., Evens, S. and Ginzburg, V.: Representations of quantum tori and double-affine Hecke algebras, math.RT/0005024.
7. Baranovsky, V. and Ginzburg, V.: Conjugacy classes in loop groups and G-bundles on elliptic curves, Internat. Math. Res. Notices (1996), No. 15, 733-751 (alg-geom/ 9607008).
8. Bouwknegt, P. and Mathai, V.: D-branes, B-fields and twisted K-theory, J. High Energy Phys. (2000), No. 3, Paper 7, 11pp. (hep-th/0002023).
9. Bondal, A. I. and Orlov, D.: Reconstruction of a variety from the derived category and groups of autoequivalences, alg-geom/97120029.
10. Cheung, Y.-K. E. and Krogh, M.: Non-commutative geometry from 0-branes in a background B-field, Nuclear Phys. B 528 (1998), 185-196 (hep-th/9803031).
11. Connes, A.: Noncommutative Geometry, Academic Press, San Diego, CA, 1994.
12. Connes, A., Douglas, M. R. and Schwarz, A.: Non-commutative geometry and Matrix theory: compactification on tori, J. High Energy Phys. (1998), No. 2, Paper 3, 35pp. (hep-th/9711162).
13. Connes, A. and Rieffel, M. A.: Yang-Mills for noncommutative two-tori, Contemp. Math. 62, Amer. Math. Soc., Providence. R.I., 1987, pp. 237-266.
14. Dijkaraaf, R.: Mirror symmetry and elliptic curves, In: The Moduli Space of Curves (Texel Island, 1994), Progr. Math. 129, Birkhäuser Boston, 1995, pp. 149-163.
15. Dijkgraaf R.: Fields, strings and duality, In: Symétries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 3-147 (hep-th/9703136).
16. Dijkgraaf, R., Verlinde, E. and Verlinde, H.: On moduli spaces of conformal field theories with c ≥ 1, In: P. Di Vecchia and J. L. Petersen (eds), Perspectives in String Theory, (Copenhagen, 1987), World Scientific, Teaneck, NJ, 1988, pp. 117-137.
17. Douglas, M. R. and Hull, C.: D-branes and the noncommutative torus, J. High Energy Phys. (1998), No. 2, Paper 8, 5 pp. (hep-th/9711165).
18. Douglas, M. R., Fiol, B. and Römelsberger, C.: Stability and BPS branes, hep-th/0002037.
19. Faddeev, L.: Modular double of a quantum group, In: G. Dito and D. Sternheimer (eds), Conférence Moshé Flato 1999, Math. Phys. Stud. 21, Kluwer Acad. Publ., Dordrecht, 2000, pp. 149-156, (math.QA/9912078).
20. Felder, G., Fröhlich, J., Fuchs, J. and Schweigert, C: Conformal boundary conditions and three-dimensional topological field theory. Phys. Rev. Lett. 84 (2000), 1659-1662 (hep-th/9909140).
21. Fukaya, K.: Floer homology of Lagrangian foliation and noncommutative mirror symmetry, Preprint 98-08, Kyoto University, 1998.
22. Fukaya, K.: Mirror symmetry of Abelian varieties and multi theta functions, Preprint Kyoto University, 1998.
23. Freed, D. S. and Witten, E.: Anomalies in strine theory with D-branes, hep-th/9907189.
24. Ginsburg, V.: Characteristic varieties and vanishing cycles, Invent. Math. 84(1986), 327-402.
25. Golyshev, V., Lunts, V. and Orlov, D.: Mirror symmetry for Abelian varieties, math.AG/9812003.
26. Gross, M. and Wilson, P.: Large complex structure limits of K3 surfaces, math.DG/0008018.
27. Kaneko, M. and Zagier, D.: A generalized Jacobi theta function and quasimodular forms, In: The Moduli Space of Curves (Texel Island, 1994), Progr. Math. 129, Birkhäuser, Boston, 1995, pp. 165-172.
28. Kapustin, A.: D-branes in a topologically nontrivial B-field, hep-th/9909089.
29. Kapustin, A., Kuznetsov, A. and Orlov, D.: Non-commutative instantons and twistor transform, hep-th/0002193.
30. Konechny, A. and Schwarz, A.: 1/4-BPS states on noncommutative tori, J. High Energy Phys. 1999). No. 9, Paper 30, 15pp. (hep-th/9907008).
31. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proc. Internat. Congress of Mathematicians, (Zürich, 1994) Vols 1, 2, Birkhäuser, Basel, 1995, pp. 120-139.
32. Kontsevich, M.: Deformation quantization of Poisson manifolds, I. q-alg/9709040.
33. Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48 (1999), 35-72 (math.QA/9904055).
34. Kontsevich, M. and Soibelman, Y.: Deformation Theory (book in preparation).
35. Kontsevich, M. and Soibelman, Y.: Homological mirror symmetry and torus fibrations, math.SG/0011041.
36. Kontsevich, M. and Soibelman, Y.: Deformations of algebras over operads and the Deligne conjecture, In: G. Dito and D. Sternheimer (eds), Conférence Moshé Flato 1999, Math. Phys. Stud. 21, Kluwer Acad. Publ., Dordrecht, 2000, pp. 255-307 (math.QA/0001151).
37. Kontsevich, M. and Rosenberg, A. L.: Non-commutative smooth spaces, In: The Gelfand Mathematical Seminars. 1996-1999, Birkhäuser, Boston, 2000, pp. 85-108 (math.AG/9812158).
38. Korogodski, L. I. and Soibelman, Y. S.: Algebras of Functions on Quantum Groups, Part I, Math. Surveys and Monogr., 56. Amer. Math. Soc., Providence, RI, 1998.
39. Landi, G., Lizzi, F. and Szabo, R. J.: String geometry and the noncommutative torus, Comm. Math. Phys. 206 (1999), 603-637 (hep-th/9806099).
40. Manin, Y. I.: Quantized theta-functions, In: Common Trends in Mathematics and Quantum Field Theories (Kyoto, 1990), Progr. Theoret. Phys. Suppl. No. 102 (1990), 219-228 (1991) (Preprint RIMS-700, 1990).
41. Manin, Y. I.: Mirror symmetry and quantization of Abelian varieties, math.AG/ 0005143.
42. Moore, G.: Finite in all directions, hep-th/9305139.
43. Orlov, D.: On equivalences of derived categories of coherent sheaves on Abelian varieties, alg-geom/9712017.
44. Orlov, D.: Equivalences of derived categories and K3 surfaces, Algebraic geometry, 7, J. Math. Sci. (N. Y.) 84 (1997), 1361-1381 (alg-geom/9606006).
45. ∞-structures on an elliptic curve, math.AG/0001048.
46. Polishchuk, A.: Homological mirror symmetry with higher products, math.AG/ 9901025.
47. Polishchuk, A. and Zaslow, E.: Categorical mirror symmetry: the elliptic curve, Adv. Theoret. Math. Phvs. 2(2) (1998), 443-470 (math.AG/9801119).
48. Rieffel M. A. and Schwarz, A.: Morita equivalence of multidimensional non-commutative tori, Internat. J. Math. 10 (1999), 289-299 (math.QA/9803057).
49. Sabbah, C.: Systémes holonomes d'équations aux q-différences. In: D-Modules and Microlocal Geometry (Lisbon, 1990), de Gruyter, Berlin, 1993, pp. 125-147.
50. Schomerus, V.: D-branes and deformation quantization, J. High Energy Phys. (1999), No. 6, Paper 30, 14 pp. (hep-th/9903205).
51. Schwarz, A.: Morita equivalence and duality, Nuclear Phys. B 534(3) (1998), 720-738 (hep-th/9805034).
52. Seiberg, N. and Witten, E.: String theory and noncommutative geometry, J. High Energy Phys. (1999), No. 9, Paper 32, 93 pp. (electronic) (hep-th/9908142).
53. Soibelman, Y.: Non-commutative compactifications and theta functions (in preparation).
54. Strominger, A., Yau, S.-T. and Zaslow, E.: Mirror symmetry is T-duality, Nuclear Phys. B 419 (1996), 243-259 (hep-th/9606040).
55. Weinstein, A.: Classical theta functions and quantum tori, Publ. Res. Inst. Math. Sci. 30 (1994), 327-333.