On brane curvature couplings from disk amplitude

The Chern–Simons coupling on $D$-branes in string theory is known to receive higher derivative corrections which include the curvature tensor and the Neveu–Schwarz $B$-field. We find some curvature couplings of $D$-branes to closed string fields by evaluating string theory disc amplitudes of one Ramond–Ramond field $C^{(p+1)}$ and two gravitons in the presence of a single $D_p$-brane in type $II$ string theory.