Notes on noncommutative supersymmetric gauge theory on the fuzzy supersphere

In these notes we review Klimč́ık’s construction of noncommutative gauge theory on the fuzzy supersphere. This theory has an exact SUSY gauge symmetry with a finite number of degrees of freedom and thus in principle it is amenable to the methods of matrix models and Monte Carlo numerical simulations. We also write down in this article a novel fuzzy supersymmetric scalar action on the fuzzy supersphere. The differential calculus on the fuzzy sphere is 3−dimensional and as a consequence a spin 1 vector field ~ C is intrinsically 3−dimensional. Each component Ci, i = 1, 2, 3, is an element of some matrix algebra MatN . Thus U(1) symmetry will be implemented by U(N) transformations. On the fuzzy sphere S N it is not possible to split the vector field ~ C in a gauge-covariant fashion into a tangent 2-dimensional gauge field and a normal scalar fluctuation. Thus in order to reduce the number of independent components from 3 to 2 we impose the gauge-covariant condition 1 2 (xiCi + Cixi) + C i √ N2 − 1 = 0. (1) xi = Li/ √ Li ( where Li are the generators of SU(2) in the irreducible representation N−1 2 of the group ) are the matrix coordinates on fuzzy S N . The action on the fuzzy sphere S 2 N is given by SN [C] = 1 4Ng2 TrF 2 ij − 1 2Ng2 ǫijkTrL [ 1 2 FijCk − i 6 [Ci, Cj]Ck ]