Central Charges in Non ( anti ) commutative N = 2 Supersymmetric U ( N ) Gauge Theory

We study the central charge of the deformed N = (1, 0) supersymmetry algebra in non(anti)commutative N = 2 supersymmetric U(N) gauge theory. In the cases of N = 1/2 superspace and N = 2 harmonic superspace with the singlet deformation, we find that the central charge is deformed by the non(anti)commutative parameters but depends on the electric and magnetic charges. For generic deformation of N = 2 harmonic superspace, we compute the O(C) correction to the central charges in the case of U(1) gauge group. Supersymmetric field theories in non(anti)commutative superspace [1, 2] arise as the low-energy effective field theories on the D-branes in the graviphoton background [3, 4, 5] and have been extensively studied in the last few years. These theories are defined in Euclidean superspace by using the ∗-product for the supercoordinates. In particular supersymmetric field theories in non(anti)commutative N = 1 superspace (N = 1/2 superspace) have been investigated both perturbatively and nonperturbatively [6, 7, 8, 9]. In the case of N = 2 superspace, there exists a variety of deformations [10, 11, 12, 13]. For generic Q-deformation, N = (1, 1) supersymmetry is deformed to N = (1, 0) [12, 13]. But for some particular deformation parameters, in which only N = 1 subspace is deformed, it is shown that the N = (1, 0) supersymmetry enhances to N = (1, 1/2) [12]. The deformed N = (1, 1/2) supersymmetry has been constructed explicitly in [14] for the U(1) gauge theory. For N = 2 supersymmetric U(N) gauge theory in non(anti)commutative N = 1 superspace, the deformed N = (1, 1/2) supersymmetry is constructed in [15]. It is interesting to study the role of the deformed supersymmetry at the quantum level. Since non(anti)commutative field theories do not have Poincaré invariance, the supersymmetry algebra could get nontrivial corrections, which avoid the HaagLopuszański-Sohnius no-go theorem [16]. In N = 1/2 supersymmetric field theories, the central charge of the deformed supersymmetry algebra was studied in [17, 18]. In particular, for N = 1/2 Wess-Zumino (WZ) model, it was shown that the formula of the central charge associated with the domain wall is not deformed but the non(anti)commutative effects enter through the deformed equations of motion[18]. In N = 2 supersymmetric gauge theory, Witten and Olive [19] have shown that there exists a central extension in N = 2 supersymmetry algebra and its central charge is related to the electric and magnetic charges of the monopoles or dyons of the theory (See [20] for reviews). Their BPS property is very important to study the strong coupling physics of the theory [21]. In this paper we will study the effects of non(anti)commutativity on the deformed N = (1, 0) supersymmetry algebra of N = 2 supersymmetric U(N) gauge theory in N = 1/2 superspace. We find that new central charges appear in the algebra but they depend on the electric and magnetic charges and the vacuum expectation value of the