SUSY anomalies break N=2 to N=1: The supersphere and the fuzzy supersphere

The N=1 SUSY on S^2 and its fuzzy finite-dimensional matrix version are known. The latter regulates quantum field theories, and seems suitable for numerical work and capable of higher dimensional generalizations. In this paper, we study their instanton sectors. They are SUSY generalizations of U(1) bundles on S^2 and their fuzzy versions, and can be characterized by $k\in\mathbb{Z}$, the SUSY Chern numbers. In the no-instanton sector (k=0), N=2 SUSY can be chirally realized, the 3 new N=2 generators anticommuting with the ``Dirac'' operator defining the free action. If $k\neq 0$, the Dirac operator has zero modes which form an N=1 supermultiplet and an atypical representation of N=2 SUSY. They break the chiral SUSY generators by the Fujikawa mechanism. We have not found this mechanism for SUSY breakdown in the literature. All these phenomena occur also on the supersphere SUSY, the graded commutative limit of the fuzzy model. We plan to discuss that as well in a later work.