  We find the terms in the nonabelian world-volume action of a system of many
Dp-branes which describe the leading coupling to all type II supergravity
background fields. These results are found by T-dualizing earlier results for
D0-branes, which in turn were determined from calculations of the M(atrix)
theory description of the supercurrent of 11D supergravity. Our results are
compatible with earlier results on the supersymmetric Born-Infeld action for a
single D-brane in a general background and with Tseytlin's symmetrized trace
proposal for extending the abelian Born-Infeld action to a nonabelian theory.
In the case p = 3, the operators we find on the D-brane world-volume are
closely related to those which couple to supergravity fields in the AdS_5 * S^5
IIB supergravity background. This gives an explicit construction, including
normalization, of some of the operators used in the celebrated AdS/CFT
correspondence for 3-branes. We also discuss the S-duality of the action in the
case p = 3, finding that the S-duality of the action determines how certain
operators in the 4D N = 4 SYM theory transform under S-duality. These S-duality
results give some new insight into the puzzle of the transverse 5-brane in
M(atrix) theory.
