Skip to main content
SpringerLink
Log in

Oscillatory Modules

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

Developing the ideas of Bressler and Soibelman and of Karabegov, we introduce a notion of an oscillatory module on a symplectic manifold which is a sheaf of modules over the sheaf of deformation quantization algebras with an additional structure. We compare the category of oscillatory modules on a torus to the Fukaya category as computed by Polishchuk and Zaslow.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. 111(1), 61–110 (1978)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Boutet de Monvel L.: Formal norms and star exponentials. Lett. Math. Phys. 83, 213–216 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Boutet de Monvel L., Guillemin V.: The Spectral Theory of Toeplitz Operators, Annals of Mathematics Studies, vol. 99. Princeton University Press, Princeton (1981)

    Google Scholar 

  4. Bressler, P., Donin, J.: Polarized deformation quantization, arXiv:math/0007186

  5. Bressler P., Gorokhovsky A., Nest R., Tsygan B.: Deformation quantization of gerbes. Adv. Math. 214(1), 230–266 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bressler, P., Gorokhovsky, A., Nest, R., Tsygan, B.: Deformations of Azumaya algebras. Actas del XVI Coloquio Latinoamericano de Álgebra, pp. 131–152. Biblioteca de la Rev. Mat. Iberoamericana (2007)

  7. Bressler, P., Gorokhovsky, A., Nest, R., Tsygan, B.: Deformations of gerbes on smooth manifolds. In: K-Theory and Noncommutative Geometry. EMS Publishing house, Switzerland (2009, in press)

  8. Bressler, P., Gorokhovsky, A., Nest, R., Tsygan, B.: Chern character for twisted complexes. In: Geometry and Dynamics of Groups and Spaces. Progress in Mathematics, vol. 265 (2008)

  9. Bressler P., Nest R., Tsygan B. Riemann–Roch theorems via deformation quantization, I, II. Adv. Math. 167(1), 1–25, 26–73 (2002)

    Google Scholar 

  10. Bressler P., Soibelman Y.: Homological mirror symmetry, deformation quantization and noncommutative geometry. J. Math. Phys. 45(10), 3972–3982 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Brylinski, J.-L.: Loop spaces, characteristic classes and geometric quantization. In: Progress in Mathematics, vol. 107. Birkhäuser, Boston (1993)

  12. Connes A.: Noncommutative Geometry, 661p. Academic Press, San Diego (1994)

    Google Scholar 

  13. Connes, A., Marcolli, M.: A walk in the noncommutative garden, arXiv:math/0601054

  14. D’Agnolo, A., Polesello, P.: Stacks of twisted modules and integral transforms. In: Geometric Aspects of Dwork Theory, vol. I, II, pp. 463–507. Walter de Gruyter GmbH & Co. KG, Berlin (2004)

  15. Deligne P.: Déformations de l’algebre des fonctions d’une variété symplectique: comparaison entre Fedosov et De Wilde, Lecomte. Sel. Math. (N.S.) 1(4), 667–697 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dito G., Schapira P.: An algebra of deformation quantization for star-exponentials on complex symplectic manifolds. Commun. Math. Phys. 273(2), 395–414 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Fang, B., Liu, C.-C.M., Treumann, D., Zaslow, E.: The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties (2009)

  18. Fang, B., Liu, C.-C.M., Treumann, D., Zaslow, E.: T-Duality and Equivariant Homological Mirror Symmetry for Toric Varieties, arXiv:0901.4276

  19. Fedosov B.V.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40, 213–238 (1994)

    MATH  MathSciNet  Google Scholar 

  20. Fedosov B.: Deformation Quantization and Index Theorem. Akademie Verlag, Berlin (1994)

    Google Scholar 

  21. Frenkel E., Witten E.: Geometric endoscopy and mirror symmetry. Commun. Number Theory Phys. 2(1), 113–283 (2008)

    MATH  MathSciNet  Google Scholar 

  22. Fock, V., Goncharov, A.: Cluster X-varieties, amalgamation, and Poisson-Lie groups. In: Algebraic Geometry and Number Theory. Progress in Mathematics, vol. 253, pp. 27–68. Birkhäuser, Boston (2006)

  23. Fukaya, K.: Deformation theory, homological algebra and mirror symmetry. In: Geometry and Physics of Branes (Como, 2001), Ser. High Energy Phys. Cosmol. Gravit., IOP, Bristol, pp. 121–209 (2003)

  24. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory—anomaly and obstruction. Kyoto preprint Math 00-17 (2000)

  25. Fuks D.: Cohomology of Infinite-Dimensional Lie Algebras. Consultants Bureau, New York (1986)

    MATH  Google Scholar 

  26. Gelfand I.M., Kazhdan D.A.: Some problems of differential geometry and the calculation of cohomologies of Lie algebras of vector fields. Sov. Math. Dokl. 12(5), 1367–1370 (1971)

    Google Scholar 

  27. Gerstenhaber M.: On the deformation of rings and algebras. Ann. Math. (2) 79, 59–103 (1964)

    Article  MathSciNet  Google Scholar 

  28. Ginzburg V.: Characteristic varieties and vanishing cycles. Invent. Math. 84(2), 327–402 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  29. Giraud J.: Cohomologie Non abélienne, Gründlehren, vol. 179. Springer, Berlin (1971)

    Google Scholar 

  30. Guillemin V.: Star products on compact pre-quantizable symplectic manifolds. Lett. Math. Phys. 35(1), 85–89 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  31. Guillemin, V., Sternberg, S.: Geometric Asymptotics, Mathematical Surveys, vol. 14. American Mathematical Society, Providence (1977)

  32. Gukov, S., Witten, E.: Branes and quantization, arXiv:hep-th/08090305

  33. Hörmander L.: Fourier integral operators I. Acta Math. 127(1–2), 79–183 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  34. Kapustin A., Witten E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1(1), 1–236 (2007)

    MathSciNet  Google Scholar 

  35. Karasev, M.: Connections on Lagrangian varieties and some quasiclassical approximation problems. In: Quantization, Coherent States, and Complex Structures, pp. 1053–1062. Plenum, New York (1992)

  36. Kashiwara M.: Quantization of contact manifolds. Publ. RIMS, Kyoto 32, 1–5 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  37. Kashiwara M., Schapira P.: Categories and Sheaves. In: Grundlehren der Mathematischen Wissenschaften, vol. 292. Springer, Berlin (1994)

    Google Scholar 

  38. Kashiwara, M., Schapira, P.: Deformation quantization modules I: Finiteness and duality, arXiv:0802.1245

  39. Kashiwara, M., Schapira, P.: Deformation quantization modules II. Hochschild class, arXiv:0809.4309

  40. Kazhdan, D.: Introduction to QFT. Quantum fields and strings: a course for mathematicians, vol. 1, 2 (Princeton, NJ, 1996/1997), pp. 377–418. American Mathematical Society, Providence (1999)

  41. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of ICM Zürich (1994)

  42. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435

  43. Leichtnam E., Nest R., Tsygan B.: Local formula for the index of a Fourier integral operator. J. Differ. Geom. 59(2), 269–300 (2001)

    MATH  MathSciNet  Google Scholar 

  44. Leray J.: Lagrangian Analysis and Quantum Mechanics. Studies in Applied Mathematics, vol. 7–9, Adv. Math. Suppl. Stud., 8. Academic Press, New York (1983)

    Google Scholar 

  45. Maslov V.: Théorie de Perturbations. Dunod, Paris (1972)

    MATH  Google Scholar 

  46. Maslov V.: Operational Methods. Mir, Moscow (1976)

    MATH  Google Scholar 

  47. Mishchenko A., Shatalov V., Sternin B.: Lagrangian Manifolds and Maslov Operator. Springer Lectures in Soviet Mathematics. Springer, Berlin (1990)

    Google Scholar 

  48. Mumford D.: Tata Lectures on Theta, I. Birkhäuser, Boston (1983)

    MATH  Google Scholar 

  49. Nadler, D.: Microlocal branes are constructible sheaves, arXiv:math/0612399

  50. Nadler D., Zaslow E.: Constructible Sheaves and the Fukaya Category. J. Am. Math. Soc. 22(1), 233–286 (2009) arXiv:math/0604379

    Article  MathSciNet  Google Scholar 

  51. Nazaikinskii V., Schulze B.-W., Sternin B.: Quantization Methods in Differential Equations, Differential and Integral Equations and Their Applications. Taylor and Francis, London (2002)

    Google Scholar 

  52. Nest R., Tsygan B.: Remarks on Modules over deformation quantization algebras. Mosc. Math. J. 4(4), 911–940 (2004)

    MATH  MathSciNet  Google Scholar 

  53. Nest R., Tsygan B.: Algebraic index theorem for families. Adv. Math. 113(2), 151–205 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  54. Polesello, P.: Classification of deformation quantization algebroids on complex symplectic manifolds, arXiv:math/0503400

  55. Polesello P., Schapira P.: Stacks of quantization-deformation modules on complex symplectic manifolds. Int. Math. Res. Not. 49, 2637–2664 (2004)

    Article  MathSciNet  Google Scholar 

  56. Polishchuk, A., Zaslow, E.: Categorical mirror symmetry in the elliptic curve. AMS/IP Stud. Adv. Math., vol. 23, pp. 275–295. AMS, Providence (1998)

  57. Schapira, P., Shneiders, J.P.: Index theorem for elliptic pairs. Astérisque, vol. 224 (1994)

  58. Seidel P.: Graded Lagrangian submanifolds. Bull. Soc. Math. France 128(1), 103–149 (2000)

    MATH  MathSciNet  Google Scholar 

  59. Tamarkin, D.: Microlocal condition for non-displaceability, arXiv:math 08091584

  60. Tamarkin, D.: Microlocal analysis and the Fukaya category of the two-torus, seminar talk, Northwestern University (2007)

  61. Weinstein, A.: Deformation Quantization, Seminaire Bourbaki, exposé 789. Astérisque, vol. 227, pp. 389–409 (1994)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Northwestern University, Evanston, IL, 60208-2730, USA

    Boris Tsygan

Authors
  1. Boris Tsygan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris Tsygan.

Additional information

In memory of Moshe Flato

This work was partially supported by NSF grant DMS-0605030.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tsygan, B. Oscillatory Modules. Lett Math Phys 88, 343–369 (2009). https://doi.org/10.1007/s11005-009-0322-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-009-0322-7

Mathematics Subject Classification (2000)

Keywords

Search

Navigation