Algebras and Representation Theory

, Volume 21, Issue 2, pp 375–397 | Cite as

Off-Shell Supersymmetry and Filtered Clifford Supermodules

  • Charles F. Doran
  • Michael G. Faux
  • Sylvester J. GatesJr.
  • Tristan Hübsch
  • Kevin Iga
  • Gregory D. Landweber


An off-shell representation of supersymmetry is a representation of the super Poincaré algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded off-shell representations of one-dimensional N-extended supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1|N}\), correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded off-shell representations of two-dimensional (p,q)-supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1,1|p,q}\), correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are Rees superalgebras and Rees supermodules, the formal deformations of filtered superalgebras and supermodules, which give a one-to-one correspondence between filtered spaces and graded spaces with even degree-shifting injections. This generalizes the machinery used by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends the notion of Rees algebras and modules to filtrations which are compatible with a supersymmetric structure. We also describe the analogous constructions for bifiltrations and bigradings.


Off-shell supersymmetry Supersymmetric quantum mechanics Super Poincaré Clifford algebra Spinor Adinkra Filtration Bifiltration 

Mathematics Subject Classification (2010)

Primary: 81Q60; Secondary: 15A66 16W70 


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C. F. Doran acknowledges support from the National Sciences and Engineering Research Council, the Pacific Institute for Mathematical Sciences, and a McCalla professorship at the University of Alberta.

S. J. Gates, Jr. would like to acknowledge that this work was partially supported by the National Science Foundation grant PHY-13515155, and that this research was also supported in part the University of Maryland Center for String and Particle Theory.


  1. 1.
    Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3(suppl. 1), 3–38 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Doran, C.F., Faux, M.G., Gates, Jr. S.J., Hübsch, T., Iga, K.M., Landweber, G.D.: On graph-theoretic identifications of Adinkras, supersymmetry representations and superfields. Internat. J. Modern Phys. A 22(5), 869–930 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Doran, C.F., Faux, M.G., Gates, Jr. S.J., Hübsch, T., Iga, K.M., Landweber, G.D.: Counter-examples to a putative classification of 1-dimensional N-extended supermultiplets. Adv. Stud. Theor. Phys. 2(1–4), 99–111 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Faux, M.G., Gates, Jr. S.J.: Adinkras: A graphical technology for supersymmetric representation theory. Phys. Rev. D71, 065002 (2005)Google Scholar
  5. 5.
    Freed, D.S., Five lectures on supersymmetry, American Mathematical Society (1999)Google Scholar
  6. 6.
    Gates, Jr. S.J., Linch, W., Phillips, J.: When superspace is not enough (2002)Google Scholar
  7. 7.
    Gates, Jr. S.J., Linch, W., Phillips, J., Rana, L.: The fundamental supersymmetry challenge remains. Gravit. Cosmol. 8(1–2), 96–100 (2002). Special issue dedicated to the centennial of Tomsk State Pedagogical UniversityMathSciNetzbMATHGoogle Scholar
  8. 8.
    Gerstenhaber, M.: On the deformation of rings and algebras II. Ann. Math. 84, 1–19 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kuznetsova, Z., Rojas, M., Toppan, F.: Classification of irreps and invariants of the N-extended supersymmetric quantum mechanics. J. High Energy Phys. 3(098), 50 (2006). electronicMathSciNetzbMATHGoogle Scholar
  10. 10.
    Lawson, H.B., Michelson, M.-L.: Spin geometry, volume 38 of Princeton Mathematical Series, Princeton University Press (1989)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Mathematical, Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Department of PhysicsSUNY College at OneontaOneontaUSA
  3. 3.Physics DepartmentUniversity of MarylandCollege ParkUSA
  4. 4.Department of PhysicsHoward UniversityWashingtonUSA
  5. 5.Department of MathematicsHoward UniversityWashingtonUSA
  6. 6.Natural Science DivisionPepperdine UniversityMalibuUSA
  7. 7.VancouverUSA

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