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A Microlocal Category Associated to a Symplectic Manifold

  • Boris TsyganEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 269)

Abstract

For a symplectic manifold subject to certain topological conditions a category enriched in \(A_{\infty }\) local systems of modules over the Novikov ring is constructed. The construction is based on the category of modules over Fedosov’s deformation quantization algebra that have an additional structure, namely an action of the fundamental groupoid up to inner automorphisms. Based in large part on the ideas of Bressler-Soibelman, Feigin, and Karabegov, it motivated by the theory of Lagrangian distributions and is related to other microlocal constructions of a category starting from a symplectic manifold, such as those due to Nadler-Zaslow and Tamarkin. In the case when our manifold is a flat two-torus, the answer is very close to both the Tamarkin microlocal category and the Fukaya category as computed by Polishchuk and Zaslow.

References

  1. 1.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys. 111, 61–151 (1977)CrossRefGoogle Scholar
  2. 2.
    Borel, A. (ed.): Algebraic \({\cal{D}}\) -Modules. Academic Press, Boston (1987)Google Scholar
  3. 3.
    Bressler, P., Soibelman, Y.: Homological mirror symmetry, deformation quantization and noncommutative geometry. J. Math. Phys. 45(10), 3972–3982 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bressler, P., Gorokhovsky, A., Nest, R., Tsygan, B.: Deformation quantization of gerbes. Adv. Math. 214(1), 230–266 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bressler, P., Gorokhovsky, A., Nest, R., Tsygan, B.: Chern character for twisted complexes. Geometry and Dynamics of Groups and Spaces. Progress in Mathematics, vol. 265, pp. 309–324. Birkhäser, Basel (2008)Google Scholar
  6. 6.
    Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press, Princeton (1956)zbMATHGoogle Scholar
  7. 7.
    Dito, G., Schapira, P.: An algebra of deformation quantization for star-exponentials on complex symplectic manifolds. Commun. Math. Phys. 273(2), 395–414 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Drinfeld, V.: DG quotients of DG categories. J. Algebr. 272(2), 643–691 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fedosov, B.: A simple geometric construction of deformation quantization. J. Differ. Geom. 40, 213–238 (1994)CrossRefGoogle Scholar
  10. 10.
    Fedosov, B.: On a spectral theorem in deformation quantization. Int. J. Geom. Methods Mod. Phys. 03, 1609 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory–anomaly and obstructions, Parts I–II. AMS/IP Studies in Advanced Mathematics, vol. 46, pp. 1–2. American Mathematical Society, Providence; International Press, Somerville (2009)Google Scholar
  12. 12.
    Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Floer theory on compact toric manifolds, I, II. Duke Math. J. 151(1), 23–174 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gillette, H.: The K-theory of twisted complexes. Contemporary Mathematics Part 1, vol. 55, pp. 159–192 (1986)Google Scholar
  14. 14.
    Gross, M., Siebert, B.: Theta functions and mirror symmetry. Surv. Differ. Geom. 21(1):95–138 (2016). arXiv:1204.1991MathSciNetCrossRefGoogle Scholar
  15. 15.
    Guillemin, V., Sternberg, S.: Geometric asymptotics. Mathematical Surveys, vol. 14. AMS, Providence (1971)Google Scholar
  16. 16.
    Hörmander, L.: Fourier integral operators I. Acta Math. 127(1–2):79–183 (1971)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Igusa, K.: Twisting cochains and higher torsion. J. Homotopy Relat. Struct. 6(2), 213–238 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1(1), 1–236 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Karabegov, A.: A formal model of Berezin deformation quantization. Commun. Math. Phys. 274, 659–689 (2007)CrossRefGoogle Scholar
  20. 20.
    Kashiwara, M.: The Riemann–Hilbert correspondence for holonomic systems. Publ. RIMS, Kyoto University 20, 319–365 (1984)CrossRefGoogle Scholar
  21. 21.
    Kashiwara, M., Schapira, P.: Sheaves on manifolds. Gründlehren der Math. Wiss., vol. 292. Springer, Berlin (1990)Google Scholar
  22. 22.
    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)Google Scholar
  23. 23.
    Keller, B.: On differential graded categories. ICM, vol. 2, pp. 151–190. European Mathematical Society, Zürich (2006)Google Scholar
  24. 24.
    Keller, B.: \(A_\infty \) algebras, modules and functor categories. Trends in Representation Theory of Algebras and Related Topics. Contemporary Mathematics, vol. 406, pp. 67–93 (2006)Google Scholar
  25. 25.
    Kontsevich, M., Soibelman, Y.: Notes on \(A_\infty \) algebras, \(A_\infty \) categories and non-commutative geometry. Homological Mirror Symmetry. Lecture Notes in Physics, vol. 757. Springer, Berlin (2009)Google Scholar
  26. 26.
    Leray, J.: Lagrangian Analysis and quantum mechanics, a mathematical structure related to the asymptotic expansions and the Maslov index. The MIT Press, Cambridge (1981). Translated from: Analyse Lagrangienne, RCP 25, Strasbourg Coll\({\grave{e}}\)ge de France (1976–1977)Google Scholar
  27. 27.
    Mackenzie, K.: Lie Groupoids and Lie Algebroids in Differential Geometry. Cambridge University Press, Cambridge (1987)CrossRefGoogle Scholar
  28. 28.
    Merkulov, S.: Strong homotopy algebras of a Kähler manifold. Int. Math. Res. Not. 3, 153–164 (1999)CrossRefGoogle Scholar
  29. 29.
    Nadler, D.: Microlocal branes are constructible sheaves. Selecta Math. (N.S.) 15(4), 65–137, 271 (2009)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. AMS 22, 19–46 (2009)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Nest, R., Tsygan, B.: Algebraic index theorem for families. Adv. Math. 113(2), 151–205MathSciNetCrossRefGoogle Scholar
  32. 32.
    Nest, R., Tsygan, B.: Remarks on modules over deformation quantization algebras. Moscow Math. J. 4(4), 911–940 (2004)MathSciNetzbMATHGoogle Scholar
  33. 33.
    O’Brian, N., Toledo, D., Tong, Y.L.: A Grothendieck–Riemann–Roch formula for maps of complex manifolds. Math. Ann. 271(4), 493–526 (1985)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Polishchuk, A., Zaslow, E.: Categorical mirror symmetry in the elliptic curve. AMS/IP Studies in Advanced Mathematics, vol. 23, pp. 275–295. American Mathematical Society, Providence (1995)Google Scholar
  35. 35.
    Positselski, L.: Two Kinds of Derived Categories, Koszul Duality, and Comodule-Contramodule Correspondence. Memoirs of the American Mathematical Society, vol. 212, 996, 133 pp. (2011)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Seidel, P.: Graded Lagrangian submanifolds. Bull. Soc. Math. France 128(1), 103–149 (2000)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Tamarkin, D.: Microlocal criterion for nondisplaceability (this volume)Google Scholar
  38. 38.
    Tamarkin, D.: Microlocal category. arXiv:1511.0896
  39. 39.
    Toledo, D., Tong, Y.L.: Duality and intersection theory in complex manifolds, I. Math. Ann. 237(1), 42–77 (1978)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Tsygan, B.: Oscillatory modules. Lett. Math. Phys. 88(1–3), 343–369 (2009)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Tsygan, B.: Noncommutative calculus and operads. Topics in Noncommutative Geometry. Clay Mathematics Proceedings, vol. 16, pp. 19–66. American Mathematical Society, Providence (2012)Google Scholar
  42. 42.
    Wei, Z.: Twisted complexes on a ringed space as a dg-enhancement of perfect complexes. arXiv:1504.05055v1
  43. 43.
    Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Northwestern UniversityEvanstonUSA

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