Non ( anti ) commutative N = 2 Supersymmetric U ( N ) Gauge Theory and Deformed Instanton Equations

We study deformed supersymmetry in N = 2 supersymmetric U(N) gauge theory in non(anti)commutative N = 1 superspace. Using the component formalism, we construct deformed N = (1, 1/2) supersymmetry explicitly. Based on the deformed supersymmetry, we discuss the C-dependence of the correlators. We also study the C-deformation of the instanton equation for the gauge group U(2). Supersymmetric field theories in non(anti)commutative superspace [1, 2] has been attracted much interests from the viewpoint of effective field theories on D-branes in the graviphoton background [3, 4, 5]. Superstrings in this background provide some interesting low-energy physics in N = 2 supersymmetric field theories [6] and their N = 1 deformations [7]. It would be important to study N = 2 supersymmetric gauge theories in non(anti)commutative superspace in order to understand graviphoton effects in the low-energy effective theories from the microscopic point of view. It is convenient to use N = 2 extended non(anti)commutative superspace for studying non(anti)commutative gauge theories where supersymmetry is manifestly realized [8, 9, 10, 11, 12, 13]. In previous papers [11, 14, 15], N = 2 supersymmetric U(1) gauge theory in non(anti)commutative N = 2 harmonic superspace has been studied. We have computed the deformed Lagrangian up to the first order in the deformation parameter C of the superspace and examined their deformed symmetries. It is, however, difficult to calculate higher order C-corrections and extend the U(1) gauge group to U(N). There exist two cases such that the deformed Lagrangian of N = 2 supersymmetric U(N) gauge theory becomes simple. One is the case of the singlet deformation where the deformation parameter belongs to the singlet representation of the R-symmetry group SU(2) [9, 10, 16, 13]. The other is the case that one introduces only deformation into N = 1 subsuperspace of N = 2 superspace. In a recent paper [15], it is shown that the O(C) Lagrangian of the U(1) theory defined in non(anti)commutative N = 2 harmonic superspace leads to the theory in the non(anti)commutative N = 1 superspace [2] by the reduction of deformation parameters and some field redefinitions. It is also shown that the theory has N = (1, 1/2) supersymmetry consistent with the Poisson structure of the theory. Here N = (1, 1/2) means that there are two chiral and one antichiral supercharges, as in [10]. In this paper, we will study N = 2 supersymmetric U(N) gauge theory in the deformed N = 1 superspace, whose Lagrangian has been constructed in [17]. We will construct deformed N = (1, 1/2) supersymmetry explicitly. Based on this symmetry, we will study the C-deformed correlators of the observables. We will also examine the the C-deformed instanton equations for the gauge group U(2). The instanton solutions in non(anti)commutative N = 1 gauge theory have been investigated in [18, 19]. In the case