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Quantum Tori, Mirror Symmetry and Deformation Theory

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Abstract

We suggest to compactify the universal covering of the moduli space of complex structures by noncommutative spaces. The latter are described by certain categories of sheaves with connections which are flat along foliations. In the case of Abelian varieties, this approach gives quantum tori as a noncommutative boundary of the moduli space. Relations to mirror symmetry, modular forms and deformation theory are discussed.

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References

  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).

    Google Scholar 

  11. Connes, A.: Noncommutative Geometry, Academic Press, San Diego, CA, 1994.

    Google Scholar 

  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.

    Google Scholar 

  14. Dijkgraaf, 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).

    Google Scholar 

  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.

    Google Scholar 

  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).

    Google Scholar 

  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 string theory with D-branes, hep-th/9907189.

  24. Ginsburg, V.: Characteristic varieties and vanishing cycles, Invent. Math. 84 (1986), 327-402.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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).

    Google Scholar 

  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).

    Google Scholar 

  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.

    Google Scholar 

  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).

    Google Scholar 

  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).

    Google Scholar 

  45. Polishchuk, A.: A ∞-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. Phys. 2(2) (1998), 443-470 (math.AG/9801119).

    Google Scholar 

  48. Rieffel, M. A. and Schwarz, A.: Morita equivalence of multidimensional non-commutative tori, Internat. J. Math. 10 (1999), 289-299 (math.QA/9803057).

    Google Scholar 

  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.

    Google Scholar 

  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).

    Google Scholar 

  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 479 (1996), 243-259 (hep-th/9606040).

    Google Scholar 

  55. Weinstein, A.: Classical theta functions and quantum tori, Publ. Res. Inst. Math. Sci. 30 (1994), 327-333.

    Google Scholar 

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  1. Department of Mathematics, Kansas State University, Manhattan, KS, 66506, U.S.A.

    Yan Soibelman

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Soibelman, Y. Quantum Tori, Mirror Symmetry and Deformation Theory. Letters in Mathematical Physics 56, 99–125 (2001). https://doi.org/10.1023/A:1010984821575

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