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Notes on A∞-Algebras, A∞-Categories and Non-Commutative Geometry

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Homological Mirror Symmetry

Part of the book series: Lecture Notes in Physics ((LNP,volume 757))

Abstract

We develop a geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. The geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from the geometric point of view. The chapter contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of the Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne’s conjecture which includes both Hochschild chains and cochains. We conclude the chapter with the description of an action of the PROP of singular chains of the topological PROP of two-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with scalar product (this action is more or less equivalent to the structure of two-dimensional Topological Field Theory associated with an “abstract” Calabi–Yau manifold).

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References

  1. M. Artin, B. Mazur, Etale Homotopy, Lect. Notes Math., 100 (1969).

    Google Scholar 

  2. A. Beilinson, V. Drinfeld, Chiral algebras (in preparation)..

    Google Scholar 

  3. A. Beilinson, V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves (in preparation).

    Google Scholar 

  4. A. Bondal, M. Kapranov, Enhanced triangulated categories, Math. USSR Sbornik, 70:1, 93–107 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Bondal, M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, math.AG/0204218.

    Google Scholar 

  6. J. Boardman, R. Vogt, Homotopy invariant algebraic structures on topological spaces, Lect. Notes Math., 347 (1973).

    Google Scholar 

  7. A. Connes, Non-commutative geometry. Academic Press, 1994.

    Google Scholar 

  8. K. Costello, Topological conformal field theories, Calabi-Yau categories and Hochschild homology, preprint (2004).

    Google Scholar 

  9. J. Cuntz, D. Quillen, Cyclic homology and non-singularity, J. Amer. Math. Soc., 8:2, 373–442 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Cuntz, D. Quillen, Algebra extensions and non-singularity, J. Amer. Math. Soc., 8:2, 251–289 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Cuntz, G. Skandalis, B. Tsygan, Cyclic homology in noncommutative geometry, Encyclopaedia of Mathematical Sciences, v. 121, Springer Verlag, p. 74–113.

    Google Scholar 

  12. P. Deligne, L. Illusie, Relevements modulo p2 et decomposition du complex de de Rham, Invent. Math. 89, 247–270 (1987).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. P. Deligne, J.S. Milne, Tannakian categories, Lect. Notes Math., 900, 101–228 (1982).

    Article  MathSciNet  Google Scholar 

  14. V. Drinfeld, D.G. quotients of DG categories, math.KT/0210114.

    Google Scholar 

  15. K. Fukaya, Y.G. Oh, H. Ohta, K. Ono, Lagrangian intersection Floer theory-anomaly and obstruction. Preprint, 2000.

    Google Scholar 

  16. E. Getzler, Cartan homotopy formulas and Gauss-Manin connection in cyclic homology, Israel Math. Conf. Proc., 7, 65–78.

    Google Scholar 

  17. V. Ginzburg, Non-commutative symplectic geometry, quiver varieties and operads, math.QA/0005165.

    Google Scholar 

  18. V. Ginzburg, Lectures on Noncommutative Geometry, \\math.AG/0506603.

    Google Scholar 

  19. V. Ginzburg, Double derivations and cyclic homology, \\math.KT/0505236.

    Google Scholar 

  20. A. Grothendieck, Technique de descente.II. Sem. Bourbaki, 195 (1959/60).

    Google Scholar 

  21. M. Kashiwara, P. Schapira, Ind-sheaves, Asterisque 271 (2001).

    Google Scholar 

  22. L.Katzarkov, M. Kontsevich, T. Pantev, Calculating Gromov-Witten invariants from Fukaya category, in preparation.

    Google Scholar 

  23. H. Kajiura, Noncommutative homotopy algebras associated with open strings, arXiv:math/0306332.

    Google Scholar 

  24. D. Kaledin, Non-commutative Cartier operator and Hodge-to-de Rham degeneration, math.AG/0511665.

    Google Scholar 

  25. R. Kaufmann,Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: Cell Operads, math.AT/0606064.

    Google Scholar 

  26. R. Kaufmann,Moduli space actions on the Hochschild co-chains of a Frobenius algebra II: Correlators, math.AT/0606065.

    Google Scholar 

  27. B. Keller, Introduction to A-infinity algebras and modules, Homology, Homotopy and Applications 3, 1–35 (2001).

    MATH  Google Scholar 

  28. B. Keller, On differential graded categories, math.KT/0601185.

    Google Scholar 

  29. M. Kontsevich, Deformation quantization of Poisson manifolds, math.QA/9709040.

    Google Scholar 

  30. M. Kontsevich, Formal non-commutative symplectic geometry. In: Gelfand Mathematical Seminars, 1990–1992, p. 173–187. Birkhävser Boston, MA, (1993).

    Google Scholar 

  31. M. Kontsevich, Feynman diagrams and low-dimensional topology. Proc. Europ. Congr. Math., 1 (1992).

    Google Scholar 

  32. M. Kontsevich, Notes on motives in finite characteristic, in preparation.

    Google Scholar 

  33. M. Kontsevich, Lecture on triangulated A-categories at Max-Planck Institut für Mathematik, (2001).

    Google Scholar 

  34. M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology and enumerativ geometry, Comm. Math. Phys., 164:3, 525–562 (1994).

    Google Scholar 

  35. M. Kontsevich, Y. Soibelman, Deformations of algebras over operads and Deligne conjecture, math.QA/0001151, published in Lett. Math. Phys., 21:1, 255–307 (2000).

    MathSciNet  Google Scholar 

  36. M. Kontsevich, Y. Soibelman, Deformation theory, (book in preparation).

    Google Scholar 

  37. L. Korogodski, Y. Soibelman, Algebras of functions on quantum groups.I. Math. Surveys Monogr. 56, AMS (1998).

    Google Scholar 

  38. L. Le Bruyn, Non-commutative geometry an n (book in preparation).

    Google Scholar 

  39. V. Lyubashenko, Category of A-categories, math.CT/0210047.

    Google Scholar 

  40. V. Lyubashenko, S. Ovsienko, A construction of A-category, math.CT/0211037.

    Google Scholar 

  41. S. Mac Lane, Categories for the working mathematician. Springer-Verlag (1971).

    Google Scholar 

  42. D. Orlov,Triangulated categories of singularities and equivalences between Landau-Ginzburg models, math.AG/0503630.

    Google Scholar 

  43. R. Rouquier, Dimensions of triangulated categories, math.CT/0310134.

    Google Scholar 

  44. S. Sanablidze, R. Umble, A diagonal of the associahedra, math. AT/0011065.

    Google Scholar 

  45. P. Seidel, Homological mirror symmetry for the quartic surface, math.SG/0310414.

    Google Scholar 

  46. Y. Soibelman, Non-commutative geometry and deformations of A∞-algebras and A∞-categories, Arbeitstagung 2003, Preprint Max-PLanck Institut für Mathematik, MPIM2003-60 h, (2003).

    Google Scholar 

  47. Y. Soibelman, Mirror symmetry and non-commutative geometry of A-categories, J. Math. Phys., 45:10, 3742–3757 (2004).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  48. D. Tamarkin, B. Tsygan, Non-commutative differential calculus, homotopy BValgebras and formality conjectures, arXiv:math/0010072.

    Google Scholar 

  49. D. Tamarkin, B. Tsygan, The ring of differential operators on forms in non-commutative calculus. Proceedings of Symposia in Pure Math., 73, 105–131 (2005).

    MathSciNet  Google Scholar 

  50. B. Toen, M. Vaquie, Moduli of objects in dg-categories, math.AG 0503269.

    Google Scholar 

  51. J. L. Verdier, Categories derivees, etat 0. Lect. Notes in Math., 569, 262–312 (1977).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. IHES, 35 route de Chartres, F-91440, France

    M. Kontsevich

  2. Department of Mathematics, KSU, Manhattan, KS 66506, USA

    Y. Soibelman

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  1. M. Kontsevich
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  2. Y. Soibelman
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Kontsevich, M., Soibelman, Y. (2008). Notes on A∞-Algebras, A∞-Categories and Non-Commutative Geometry. In: Homological Mirror Symmetry. Lecture Notes in Physics, vol 757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68030-7_6

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  • DOI: https://doi.org/10.1007/978-3-540-68030-7_6

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-68029-1

  • Online ISBN: 978-3-540-68030-7

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

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