Skip to main content

BFV-Complex and Higher Homotopy Structures

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.

This is a preview of subscription content, access via your institution.

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

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Florian Schätz.

Additional information

Communicated by N. A. Nekrasov

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

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