The full low energy effective action of N = 4 SYM is believed to be self-dual. Starting with the first two leading terms in a momentum expansion of this effective action, we perform a duality transformation and find the conditions for self-duality. These determine some of the higher order terms. We compare the effective action of N = 4 SYM with the probe-source description of type II B D3-branes in the AdS 5 × S 5 background. We find agreement up to six derivative terms if we identify the separation of the 3-branes with a redefinition of the gauge scalar that involves the gauge field strength.1

We present a proposal for the description of the Higgs branch of four-dimensional N =2 supersymmetric Yang-Mills theories in the context of the AdS/CFT correspondence. We focus on a finite Sp(N ) N =2 theory arising as dual of a configuration of N D3-branes in the vicinity of four D7-branes and an orientifold 7-plane in type I ′ string theory. The field theory contains hypermultiplets in the second rank anti-symmetric and in the fundamental representations. The Higgs branch has a dual description in terms of gauge field configurations with non-zero instanton number on the world-volume of the D7-branes. In this setting the non-renormalisation of the metric on the Higgs branch implies constraints on the α ′ corrections to the D7-brane effective action, including couplings to the curvature and five-form field strength. In the second part of the paper we discuss non-renormalisation properties of BPS Wilson lines, which are closely related to the physics of the Higgs branch. Using a formulation of the four-dimensional N =2 theory in terms of a three-dimensional N =2 superspace we show that the expectation value of certain Wilson-line operators with hypermultiplets at the end points is independent of the length and thus coincides with the expectation value of the local operators parametrising the Higgs

Certain two and three point functions of gauge invariant primary operators of N =4 SYM are computed in N =1 superspace keeping all the θ-components. This allows one to read off many component descendent correlators. Our results show the only possible g 2 Y M corrections to the free field correlators are contact terms. Therefore they vanish for operators at separate points, verifying the known non-renormalization theorems. This also implies the results are consistent with N =4 supersymmetry even though the Lagrangian we use has only N =1 manifest supersymmetry. We repeat some of the calculations using supersymmetric Landau gauge and obtain, as expected, the same results as those of supersymmetric Feynman gauge.1 4 J J Figure 9:The minus sign of the graph in the left hand side of the first equality comes from the color factor. The second equality can be seen easily by writing out both sides explicitly. Finally the Fig. 1c can be written -4 J J O = 4 J J