Singlet Deformation and Non ( anti ) commutative N = 2 Supersymmetric U ( 1 ) Gauge Theory

We study N = 2 supersymmetric U(1) gauge theory in non(anti)commutative N = 2 harmonic superspace with the singlet deformation, which preserves chirality. We construct a Lagrangian which is invariant under both the deformed gauge and supersymmetry transformations. We find the field redefinition such that the N = 2 vector multilplet transforms canonically under the deformed symmetries. Non(anti)commutative superspace[1] with nonanticommutativity in Grassmann odd coordinates appears in superstrings compactified on Calabi-Yau threefold in the graviphoton background [2, 3, 4]. The low-energy effective theory on the D-brane is realized by supersymmetric gauge theories in non(anti)commutative superspace. Perturabative and non-perturabative aspects of these gauge theories have been studied extensively[5, 6, 7, 8]. It is an interesting problem to study the deformation of extended superspace since it admits a variety of deformation parameters[9]. The deformation of extended superspace has been recently studied in [10, 11, 12, 13, 14, 15]. In a previous paper [12], we have studied the deformed Lagrangian explicitly in the component formalism up to the first order in the deformation parameter C. Since the Lagrangian get higher order correction in C, the full Lagrangian is rather complicated. Moreover, the harmonic superspace formalism introduces the infinite number of auxiliary fields. In order to preserve the WZ gauge in the deformed theory, the gauge transformation has also correction in the form of power series in C. There exist some interesting cases where the deformation structure becomes simple. One is the limit to the N = 1/2 superspace[5], where the action should reduce to N = 1/2 super Yang-Mills theory with adjoint matter. Another interesting case is the singlet deformation[10, 11], where the deformation parameters belongs to the singlet representation of theR-symmetry group SU(2)R. In this paper, we will studyN = 2 supersymmetric U(1) gauge theory in the harmonic superspace with singlet deformation. In this case, the gauge and supersymmetry transformations get correction linear in the deformation parameter. Therefore we can easily perform the field redefinition such that the component fields transform canonically under the gauge transformation. In the case of N = 1/2 super Yang-Mills theory, such field redefinition is also possible[5]. But in this case the component fields do not transform canonically under the deformed supersymmtery transformation. In the singlet case, we will show that there is a field redefinition such that the redefined fields also transform canonically under the deformed supersymmetry. We will construct a deformed Lagrangian which is invariant under both the gauge and supersymmetry transformations. We find that the deformed Lagrangian is characterized by a single function of an anti-holomorphic scalar field. We begin with reviewing the non(anti)commutative deformation of N = 2 harmonic