arXiv:hep-th/0306226v2 30 Jun 2003 hep-th/0306226 Superstrings in Graviphoton Background and N = 1 2 + 3 2 Supersymmetry Nathan Berkovits Instituto de F´ ısica Te´ orica, Universidade Estadual Paulista R. Pamplona 145, S˜ ao Paulo, SP 01405-900 BRASIL Nathan Seiberg School of Natural Sciences, Institute for Advanced Study Princeton NJ 08540 USA Motivated by Ooguri and Vafa, we study superstrings in flat IR 4 in a constant self-dual graviphoton background. The supergravity equations of motion are satisfied in this back- ground which deforms the N =2 d = 4 flat space super-Poincar´ e algebra to another algebra with eight supercharges. A D-brane in this space preserves a quarter of the supercharges; i.e. N = 1 2 supersymmetry is realized linearly, and the remaining N = 3 2 supersymmetry is realized nonlinearly. The theory on the brane can be described as a theory in noncommu- tative superspace in which the chiral fermionic coordinates θ α of N =1 d = 4 superspace are not Grassman variables but satisfy a Clifford algebra. June 2003