S-duality of D-brane action at order O(α′2)

Using the compatibility of the DBI and the Chern-Simons actions with the T-duality transformations, the curvature corrections to these actions have been recently extended to include the quadratic B-field couplings at order O(α′2). In this paper, we use the compatibility of the couplings on D3-brane with the S-duality to find the nonlinear RR couplings at order O(α′2). We confirm the quadratic RR couplings in the DBI part with the disk-level scattering amplitude. Using the modular functions that appear in the curvature corrections, we then write the results in SL(2,Z) invariant form.