Abstract
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.
Similar content being viewed by others
References
Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3(suppl. 1), 3–38 (1964)
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)
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)
Faux, M.G., Gates, Jr. S.J.: Adinkras: A graphical technology for supersymmetric representation theory. Phys. Rev. D71, 065002 (2005)
Freed, D.S., Five lectures on supersymmetry, American Mathematical Society (1999)
Gates, Jr. S.J., Linch, W., Phillips, J.: When superspace is not enough (2002)
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 University
Gerstenhaber, M.: On the deformation of rings and algebras II. Ann. Math. 84, 1–19 (1966)
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). electronic
Lawson, H.B., Michelson, M.-L.: Spin geometry, volume 38 of Princeton Mathematical Series, Princeton University Press (1989)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Peter Littelmann.
Rights and permissions
About this article
Cite this article
Doran, C.F., Faux, M.G., Gates, S.J. et al. Off-Shell Supersymmetry and Filtered Clifford Supermodules. Algebr Represent Theor 21, 375–397 (2018). https://doi.org/10.1007/s10468-017-9718-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-017-9718-8
Keywords
- Off-shell supersymmetry
- Supersymmetric quantum mechanics
- Super Poincaré
- Clifford algebra
- Spinor
- Adinkra
- Filtration
- Bifiltration