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# Clifford Algebras, Supersymmetry and \$Z_n-\$symmetries: Applications in Field Theory

After a short introduction on Clifford algebras of polynomials, we give a general method of constructing a matrix representation. This process of linearization leads naturally to two fundamental structures: the generalized Clifford algebra (GCA) and the generalized Grassmann algebra (GGA) which are studied. Then, it is proved that if we equip the GGA with a differential structure, we obtain the \$q-\$deformed Heisenberg algebra or the \$q-\$oscillators. Finally, it is shown that the \$q-\$deformed Heisenberg algebra is the basic tool to define an adapted superspace leading to the extension of supersymmetry called fractional supersymmetry of order \$F\$ (FSUSY), \$F=2\$ corresponding to the usual supersymmetry. Local FSUSY in one dimension is then contructed in the world-line formalism, and an extension of the Dirac equation is obtained. In two dimensions, it turns out that FSUSY is a conformal field theory and in addition to the stress energy tensor, a supercurrent of conformal weight \$1+1/F\$, which generates a symmetry between the primary fields of conformal weight \$(0,1, cdots, 1-1/F\$), is obtained. The algebra is explicitly constructed. We also show that in \$1+2\$ dimensions FSUSY is a non-trivial extension of the Poincar'e algebra which generates a symmetry among fractional spin states or anyons. Unitarity of the representation is checked. Finally, we prove that, independently of the dimension, a natural classification emerges according to the decomposition of \$F\$ as a product of prime numbers and that FSUSY is a symmetry which closes non-linearly, and is sustained by mathematical structures that go beyond Lie or super-Lie algebras.

 Comments: LaTex, fancyheadings.sty, epsfig.sty, floatfig.sty, 106 pages, 2 figures, Habilitation Thesis (in French) Subjects: High Energy Physics - Theory (hep-th) Report number: LPT Strasbourg 97-02 Cite as: arXiv:hep-th/9802141v1

## Submission history

From: Michel Rausch [view email
[v1] Fri, 20 Feb 1998 10:35:31 GMT (100kb)

12:19 Publié dans Clifford Algebras, Supersymmetry | Lien permanent | Commentaires (0) | |  del.icio.us | |  Digg |  Facebook