In this paper, using a Hopf-algebraic method, we construct deformed Poincaré SUSY algebra in terms of twisted (Hopf) algebra. By adapting this twist deformed superPoincaré algrebra as our fundamental symmetry, we can see the consistency between the algebra and non(anti)commutative relation among (super)coordinates and interpret that symmetry of non(anti)commutative QFT is in fact twisted one. The key point is validity of our new twist element that guarantees non(anti)commutativity of space. It is checked in this paper for N = 1 case. We also comment on the possibility of noncommutative central charge coordinate. Finally, because our twist operation does not break the original algebra, we can claim that (twisted) SUSY is not broken in contrast to the string inspired N = 1/2 SUSY in N = 1 non(anti)commutative superspace.

We study deformation of N = 4 super Yang-Mills theory from type IIB superstrings with D3-branes in the constant R-R background. We compute disk amplitudes with one graviphoton vertex operator and investigate the zero-slope limit of the amplitudes. We obtain the effective action deformed by the graviphoton background, which contains the one defined in non(anti)commutative N = 1 superspace as special case. The bosonic part of the Lagrangian gives the Chern-Simons term coupled with the R-R potential. We study the vacuum configuration of the deformed Lagrangian and find the fuzzy sphere configuration for scalar fields.

We studied a nilpotent Non-Anti-Commutative (NAC) deformation of the effective superpotentials in supersymmetric gauge theories, caused by a constant self-dual graviphoton background. We derived the simple non-perturbative formula applicable to any NAC (star) deformed chiral superpotential. It is remarkable that the deformed superpotential is always 'Lorentz'-invariant. As an application, we considered the NAC deformation of the pure super-Yang-Mills theory whose IR physics is known to be described by the Veneziano-Yankielowicz superpotential (in the undeformed case). The unbroken gauge invariance of the deformed effective action gives rise to severe restrictions on its form. We found a non-vanishing gluino condensate in vacuum but no further dynamical supersymmetry breaking in the deformed theory.

We investigate deformed superconformal symmetries on non(anti)commutative (super)spaces from the point of view of the Drinfel'd twisted symmetries. We classify all possible twist elements derived from an abelian subsector of the superconformal algebra. The symmetry breaking caused by the non(anti)commutativity of the (super)spaces is naturally interpreted as the modification of their coproduct emerging from the corresponding twist element. The remaining unbroken symmetries are determined by the commutative properties of those symmetry generators possessing the twist element. We also comment on non-canonically deformed non(anti)commutative superspaces, particularly those derived from the superconformal twist element F SS . §1. IntroductionThe study of noncommutative spaces has recently attracted considerable interests, because it is thought that they may provide a fundamental basis for a theory of quantum gravity. 4) Superstring theory, which is believed to be the most promising possibility as a consistent theory of quantum gravity, provides a realization of noncommutative space. 3) The simplest noncommutative space, a noncommutative plane, possesses a so-called canonical structure among its coordinates expressed aswhere θ mn is a constant noncommutativity parameter. Note that this canonical noncommutativity breaks the Lorentz invariance of the theory. Field theories on such a noncommutative plane have been intensively studied. (See for example Ref. 4) and references therein.) The space-time noncommutative plane has been generalized to superspaces. 5), 6) The supersymmetric counterpart of Eq. (1) is given by {θ α , θ β } = C αβ = 0,(2) * )

We investigate the theory of the bosonic-fermionic noncommutativity, x ; i , and the WessZumino model deformed by the noncommutativity. Such noncommutativity links well-known space-time noncommutativity to superspace non-anticommutativity. The deformation has the nilpotency. We can explicitly evaluate noncommutative effect in terms of new interactions between component fields. The interaction terms that have Grassmann couplings are induced. The noncommutativity does completely break full N 1 supersymmetry to N 0 theory in Minkowski signature. Similar to the space-time noncommutativity, this theory has higher derivative terms and becomes nonlocal theory. However this nonlocality is milder than the space-time noncommutative field theory. Because of the nilpotent feature of the coupling constants, we find that there are only finite number of Feynman diagrams that give noncommutative corrections at each loop order.