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BFV-Complex and Higher Homotopy Structures

Communications in Mathematical Physics volume 286, Article number: 399 (2009) Cite this article

Abstract

We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are L ∞ quasi-isomorphic and control the same formal deformation problem.

However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead.

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References

  1. Batalin I.A., Fradkin E.S.: A generalized canonical formalism and quantization of reducible gauge theories. Phys. Lett. 122, 157–164 (1983)

    MATH  MathSciNet  Google Scholar 

  2. Batalin I.A., Vilkovisky G.S.: Relativistic S-matrix of dynamical systems with bosons and fermion constraints. Phys. Lett. 69, 309–312 (1977)

    Google Scholar 

  3. Bordemann, M.: The deformation quantization of certain super-Poisson brackets and BRST cohomology. http://arXiv.org/list/math.QA/0003218, 2000

  4. Cattaneo, A.S.: Deformation Quantization and Reduction. In: Poisson Geometry in Mathematics and Physics, eds. G. Dito, J.-H. Lu, Y. Maeda, A. Weinstein, Cont. Math. 450, Providence, RI: Amer. Math. Soc., 2008, pp. 79–101

  5. Cattaneo A.S., Felder G.: Relative formality theorem and quantisation of coisotropic submanifolds. Adv. Math. 208, 521–548 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  6. Cattaneo A.S., Schätz F.: Equivalences of higher derived brackets. J. Pure Appl. Algebra 212(11), 2450–2460 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  7. Costello, K.: Renormalization in the BV-Formalism. http://arXiv.org/absmath/0706.1533v3[math.QA], 2007

  8. Gotay M.: On coisotropic imbeddings of pre-symplectic manifolds. Proc. Amer. Math. Soc. 84, 111–114 (1982)

    MATH  Article  MathSciNet  Google Scholar 

  9. Gugenheim, A.K.A.M., Lambe, L.A.: Perturbation theory in differential homological algebra I. Il. J. Math. 33 (1989)

  10. Herbig, H.-C.: Variations on homological Reduction. Ph.D. Thesis University of Frankfurt, http://arXiv.org/abs/0708.3598v1[math.QA], 2004

  11. Kajiura H.: Noncommutative homotopy algebras associated with open strings. Rev. Math. Phys. 19, 1–99 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  12. Kieserman, N.: The Liouville phenomenon in the deformation problem of coisotropics. Preprint, http://arXiv.org/abs/0805.2468v1[math.GT], 2008

  13. Kontsevich, M., Soibelman, Y.: Homological mirror Symmetry and torus fibrations. http://arXiv.org/abs/math.SG/0011041v2[math.SG], 2001

  14. Lada T., Stasheff J.: Introduction to sh Lie algebras for physicists. Int. J. Theor. Phys. 32, 1087–1104 (1993)

    MATH  Article  MathSciNet  Google Scholar 

  15. Merkulov S.A.: Strongly homotopy algebras of a Kähler manifold. Internat. Math. Res. Notices 1999(3), 153–164 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  16. Mnëv, P.: Notes on simplicial BF theory. http://arXiv.org/abs/hep-th/0610326v3, 2007

  17. Oh Y.G., Park J.S.: Deformations of coisotropic submanifolds and strong homotopy Lie algebroids. Invent. Math. 161, 287–36 (2005)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  18. Rothstein, M.: The structure of supersymplectic supermanifolds. In: Differential Geometric Methods in Mathematical Physics. eds. C. Barecci et al., Proc. 19th Int. Conf., Rapallo/Italy 1990, Lect. Notes Phys. 375, Berlin-Heidelberg: Springer, 1991, pp. 331–343

  19. Schätz, F.: Invariance of the BFV-complex. In preparation

  20. Schwarz A.: Geometry of Batalin–Vilkovisky quantization. Commun. Math. Phys. 155, 249–260 (1993)

    MATH  Article  ADS  Google Scholar 

  21. Stasheff J.: The intrinsic bracket on the deformation complex of an associative algebra. J. Pure Appl. Algebra 89, 231–235 (1993)

    MATH  Article  MathSciNet  Google Scholar 

  22. Stasheff J.: Homological reduction of constrained Poisson algebras. J. Diff. Geom. 45, 221–240 (1997)

    MATH  MathSciNet  Google Scholar 

  23. Stefan P.: Accessible sets, orbits, and foliations with singularities. Proc. London Math. Soc. 20, 699–713 (1974)

    Article  MathSciNet  Google Scholar 

  24. Sussmann H.J.: Orbit of families of vector fields and integrability of distributions. Trans. Amer. Math. J. 180, 171–188 (1973)

    MATH  Article  MathSciNet  Google Scholar 

  25. Voronov Th.: Higher derived brackets and homotopy algebras. J. Pure Appl. Algebra 202(1–3), 133–153 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  26. Weinstein A.: Symplectic manifolds and their Lagrangian submanifolds. Adv. Math. 6, 329–346 (1971)

    MATH  Article  Google Scholar 

  27. Weinstein A.: Coisotropic calculus and Poisson groupoids. J. Math. Soc. Japan 40, 705–727 (1988)

    MATH  Article  MathSciNet  Google Scholar 

  28. Zambon, M.: Averaging techniques in Riemannian, symplectic and contact geometry. Ph.D. Thesis, University of Berkeley

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Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich–Irchel, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland

    Florian Schätz

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  1. Florian Schätz
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Correspondence to Florian Schätz.

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Communicated by N. A. Nekrasov

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Schätz, F. BFV-Complex and Higher Homotopy Structures. Commun. Math. Phys. 286, 399 (2009). https://doi.org/10.1007/s00220-008-0705-0

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  • DOI: https://doi.org/10.1007/s00220-008-0705-0

Keywords

  • Vector Bundle
  • Poisson Algebra
  • Poisson Manifold
  • Grade Vector Space
  • Trivalent Vertex