  More evidence is provided for the conjectured correspondence between the
D3-brane action in AdS_5 x S^5 and the low-energy effective action for N = 4
SU(N) SYM on its Coulomb branch, where the gauge group SU(N) is spontaneously
broken to SU(N-1) x U(1) and the dynamics is described by a single N = 2 vector
multiplet corresponding to the U(1) factor of the unbroken group. Using an
off-shell formulation for N = 4 SYM in N = 2 harmonic superspace, within the
background-field quantization scheme we compute the two-loop quantum correction
to a holomorphic sector of the effective action, which is a supersymmetric
completion of interactions of the form \Omega ((F^+)^2 |Y|^{-4}) (F^+)^2(F^-)^2
|Y|^{-4}, with F^\pm the (anti) self-dual components of the U(1) gauge field
strength, and Y the complex scalar belonging to the vector multiplet. In the
one-loop approximation, \Omega was shown in hep-th/9911221 to be constant. It
is demonstrated in the present paper that \Omega \propto (F^+)^2 |Y|^{-4} at
the two-loop order. The corresponding coefficient proves to agree with the F^6
coefficient in the D3-brane action, after implementing the nonlinear field
redefinition which was sketched in hep-th/9810152 and which relates the N = 2
vector multiplet component fields with those living on the D3-brane. In the
approximation considered, our results are consistent with the conjecture of
hep-th/9810152 that the N = 4 SYM effective action is self-dual under N = 2
superfield Legendre transformation, and also with the stronger conjecture of
hep-th/0001068 that it is self-dual under supersymmetric U(1) duality
rotations.
