Lorentz invariant supersymmetric mechanism for non(anti)commutative deformations of space-t.pdf

a r X i v : h e p - t h / 0 5 0 6 1 2 7 v 3 1 9 J u n 2 0 0 5 Lorentz invariant supersymmetric mechanism for non(anti)commutative deformations of space-time geometry A.A. Zheltukhina,b a Kharkov Institute of Physics and Technology, 61108 Kharkov, Ukraine b Institute of Theoretical Physics, University of Stockholm SE-10691, AlbaNova, Stockholm, Sweden Abstract A supersymmetric Lorentz invariant mechanism for superspace deformations is pro- posed. It is based on an extension of superspace by one λa or several Majorana spinors associated with the Penrose twistor picture. Some examples of Lorentz invariant su- persymmetric Poisson and Moyal brackets are constructed and the correspondence: θmn ? iψmψn, Cab ? λaλb, Ψ a m ? ψmλ a mapping the brackets depending on the constant background into the Lorentz covariant supersymmetric brackets is estab- lished. The correspondence reveals the role of the composite anticommuting vector ψm = ? 1 2(θ?γmλ) as a covariant measure of space-time coordinate noncommutativity. 1 Introduction The unification of physics and mathematics in the development of noncommutative quantum geometry [1, 2, 3, 4, 5, 6, 7] and field theories [8, 9, 10, 11, 12, 13] resulted in new ideas and approaches (see reviews [14], [15] and additional references therein). One of them has come from string theory, where the noncommutativity of the bosonic string coordinates xm in the presence of the constant antisymmetric field Bmn was observed [10]. More recently, the non- commutativity between the components of the odd spinor coordinate θa in the presence of a constant graviphoton field Cab was considered in [16]. The constant gravitino background Ψ a m resulted in the noncommutativity between the xm and θa coordinates [17, 18]. These results have focused attention on the role of constant background fields in superspace deformations. Studying field/string theories and supersymmetry preservation in the superspaces deformed by the graviphoton background [17], [18]