Maximal Supersymmetry

  • M. V. Movshev
  • A. Schwarz
Part of the Lecture Notes in Mathematics book series (LNM, volume 2027)


We have studied supersymmetric and super Poincaré invariant deformations of maximally supersymmetric gauge theories, in particular, of ten-dimensional super Yang-Mills theory and of its reduction to a point. We have described all infinitesimal super Poincaré invariant deformations of equations of motion and proved that all of them are Lagrangian deformations and all of them can be extended to formal deformations. Our methods are based on homological algebra, in particular, on the theory of L-infinity and A-infinity algebras. In this paper we formulate some of the results we have obtained, but skip all proofs. However, we describe (in Sects. 2 and 3) the results of the theory of L-infinity and A-infinity algebras that serve as the main tool in our calculations.


Formal Power Series Ghost Number Differential Algebra Hochschild Cohomology Cyclic Homology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The work of both authors was partially supported by NSF grant No. DMS 0505735 and by grants DE-FG02-90ER40542 and PHY99-0794.


  1. 1.
    N. Berkovits, Explaining the pure spinor formalism for the superstring. JHEP0801:065, arXiv:0712.0324 (2008)Google Scholar
  2. 2.
    D. Burghelea, Z. Fiedorowicz, W. Gajda, Adams operations in Hochschild and cyclic homology of de Rham algebra and free loop spaces. K-Theory 4(3), 269–287 (1991)Google Scholar
  3. 3.
    B.L. Feigin, B.L. Tsygan, in Cyclic Homology of Algebras with Quadratic Relations, Universal Enveloping Algebras and Group Algebras. K-Theory, Arithmetic and Geometry (Moscow, 1984–1986). Lecture Notes in Mathematics, vol. 1289 (Springer, Berlin, 1987), pp. 210–239Google Scholar
  4. 4.
    B. Keller, Invariance and localization for cyclic homology of DG algebras. J. Pure Appl. Algebra 123(1–3), 223–273 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    B. Keller, Derived invariance of higher structures on the Hochschild complex. Preprint available at
  6. 6.
    M. Kontsevich, private communicationGoogle Scholar
  7. 7.
    J.-L. Loday, in Cyclic Homology. Grundlehren der Mathematischen Wissenschaften, vol. 301 (Springer, Berlin, 1998)Google Scholar
  8. 8.
    M. Movshev, Cohomology of Yang-Mills algebras. J. Noncommut. Geom. 2(3), 353–404 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    M. Movshev, Deformation of maximally supersymmetric Yang-Mills theory in dimensions 10. An algebraic approach. arXiv:hep-th/0601010v1Google Scholar
  10. 10.
    M. Movshev, Deformation of maximally supersymmetric Yang-Mills theory in dimensions 10. An algebraic approach. hep-th/0601010Google Scholar
  11. 11.
    M. Movshev, Yang-Mills theories in dimensions 3, 4, 6, 10 and Bar-duality. hep-th/0503165Google Scholar
  12. 12.
    M. Movshev, A. Schwarz, Algebraic structure of Yang-Mills theory. hep-th/0404183Google Scholar
  13. 13.
    M. Movshev, A. Schwarz, On maximally supersymmetric Yang-Mills theories. hep-th/0311132Google Scholar
  14. 14.
    M. Movshev, A. Schwarz, Supersymmetric Deformations of Maximally Supersymmetric Gauge Theories. IGoogle Scholar
  15. 15.
    M. Penkava, A. Schwarz, A algebras and the cohomology of moduli spaces. Am. Math. Soc. Transl. Ser. 2, 169 (1995). hep-th/9408064Google Scholar
  16. 16.
    A. Schwarz, Semiclassical approximation in Batalin-Vilkovisky formalism. Commun. Math. Phys. 158, 373–396 (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • M. V. Movshev
    • 1
  • A. Schwarz
    • 2
  1. 1.Stony Brook UniversityStony BrookUSA
  2. 2.Department of MathematicsUniversity of CaliforniaDavisUSA

Personalised recommendations