D-brane action at order ${\alpha }^{\prime 2}$

We use the compatibility of D-brane action with linear T-duality, S-duality, and with S-matrix elements as guiding principles to find all world volume couplings of one massless closed string and two open strings at order ${\alpha }^{\prime 2}$ in type-II superstring theories. In particular, we find that the squares of second fundamental form appear only in world volume curvatures and confirm the observation that the dilaton appears in the string frame action via the transformation ${\widehat{R}}_{\mu \nu }\to {\widehat{R}}_{\mu \nu }+{\nabla }_{\mu }{\nabla }_{\nu }\mathrm{\Phi }$.