Skip to main content
SpringerLink
Log in

Matrix De Rham Complex and Quantum A-infinity algebras

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

Abstract

I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A ∞-algebras, introduced in Barannikov (Modular operads and non-commutative Batalin–Vilkovisky geometry. IMRN, vol. 2007, rnm075. Max Planck Institute for Mathematics 2006–48, 2007), is represented via de Rham differential acting on the supermatrix spaces related with Bernstein–Leites simple associative algebras with odd trace q(N), and gl(N|N). I also show that the matrix Lagrangians from Barannikov (Noncommutative Batalin–Vilkovisky geometry and matrix integrals. Isaac Newton Institute for Mathematical Sciences, Cambridge University, 2006) are represented by equivariantly closed differential forms.

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. Alexandrov M., Schwarz A., Zaboronsky O., Kontsevich M.: The geometry of the master equation and topological quantum field theory. Intern. J. Mod. Phys. A 12(7), 1405–1429 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Barannikov, S.: Quantum periods—I. Semi-infinite variations of Hodge structures. Preprint ENS DMA-00-19. Intern. Math. Res. Notices 23 (2001)

  3. Barannikov, S.: Modular operads and non-commutative Batalin–Vilkovisky geometry. IMRN, vol. 2007, rnm075. Preprint (2007). Max Planck Institute for Mathematics 2006–48 (25/04/2006)

  4. Barannikov, S.: Noncommutative Batalin–Vilkovisky geometry and matrix integrals. Preprint NI06043 (2006). Isaac Newton Institute for Mathematical Sciences, Cambridge University. Preprint hal-00102085 (09/2006). Comptes Rendus Mathématique, of the French Academy of Sciences, submitted on May 17 2009, vol. 348, issue no. 7–8, pp. 359–362

  5. Barannikov, S.: Supersymmetry and cohomology of graph complexes. Preprint hal-00429963 (2009)

  6. Barannikov, S.: Supersymmetric matrix integrals and sigma-model. Preprint hal-00443592 (2009)

  7. Barannikov, S.: (2013)

  8. Bernstein J.N., Leites D.A.: The superalgebra Q(n), the odd trace, and the odd determinant. Dokl. Bolg. Akad. Nauk 35(3), 285–286 (1982)

    MATH  MathSciNet  Google Scholar 

  9. Feigin, B.,Tsygan, B.:Additive K-theory. Springer, LNM, vol. 1289, pp. 97–209 (1987)

  10. Getzler E., Kapranov M.: Modular operads. Compos. Math. 110(1), 65–126 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Goodman R., Wallach N.R.: Representations and invariants of the classical groups. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  12. Kontsevich, M.: Formal (non)commutative symplectic geometry. The Gelfand Mathematical Seminars, vol. 1990–1992, pp. 173–187. Birkhauser, Boston (1993)

  13. Loday J.-L.: Cyclic homology. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  14. Sergeev, A.: Tensornaya algebra tozhdestvennogo predstavleniya kak modul nad superalgebrami Lie gl(n,m) i q(n). Mat. Sbornik 123(165) v.3, 422–430 (1984)

Download references

Author information

Authors and Affiliations

  1. Ecole Normale Superieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France

    S. Barannikov

Authors
  1. S. Barannikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Barannikov.

Additional information

Submitted for publication on 20/01/2010, preprint HAL-00378776 (04/2009).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barannikov, S. Matrix De Rham Complex and Quantum A-infinity algebras. Lett Math Phys 104, 373–395 (2014). https://doi.org/10.1007/s11005-013-0677-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-013-0677-7

Mathematics Subject Classification (2010)

Keywords

Search

Navigation