Two-loop N=4 super-Yang-Mills effective action and interaction between D3-branes

We compute the leading low-energy term in the planar part of the 2-loop contribution to the effective action of N = 4 SYM theory in 4 dimensions, assuming that the gauge group SU(N + 1) is broken to SU(N) x U(1) by a constant scalar background X. While the leading 1-loop correction is the familiar c(1)F(4)/\X\(4) term, the 2-loop expression starts with c(2)F(6)/\X\(8). The 1-loop constant c(1) is known to be equal to the coefficient of the F-4 term in the Born-Infeld action for a probe D3-brane separated by distance \X\ from a large number N of coincident D3-branes. We show that the sa-me is true also for the 2-loop constant c(2): it matches the coefficient of the F-6 term in the D3-brane probe action. In the context of the AdS/CFT correspondence. this agreement suggests a non-renormalization of the coefficient of the F-6 term beyond two loops. Thus the result of hepth/9706072 about the agreement between the nu(6) term in the D0-brane supergravity interaction potential and the corresponding 2-loop term in the (1 + 0)-dimensional reduction of M = 4 SYM theory has indeed a direct generalization to 1 + 3 dimensions, as conjectured earlier in hepth/9709087. We also discuss the issue of gauge theory-supergravity correspondence for higher order (F-8, etc.) terms. (C) 2002 Elsevier Science B.V. All rights reserved.