Couplings of order six in the gauge field strength and the second fundamental form on a D${}_p$-brane at order ${\alpha }^{\mathrm{\prime }2}$

Using the assumption that the independent gauge invariant couplings on the world-volume of the non-perturbative objects in the string theory are independent of the background, we find the four and the six gauge field strength and/or the second fundamental form couplings on the world volume of a D${}_p$-brane in the superstring theory at order ${\alpha }^{\mathrm{\prime }2}$ in the normalization that $F$ is dimensionless. We have found them by considering the particular background which has one circle and by imposing the corresponding T-duality constraint on the independent couplings. In particular, we find that there are 12+146 independent gauge invariant couplings at this order, and the T-duality constraint can fix 150 of them. We show that these couplings are fully consistent with the partial results in the literature. This comparison also fixes the remaining 8 couplings.