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Using the assumption that the independent gauge invariant couplings on theworld-volume of the non-perturbative objects in the string theory areindependent of the background, we find the four and the six gauge fieldstrength and/or the second fundamental form couplings on the world volume of aD$_p$-brane in the superstring theory at order $\alpha'^2$ in the normalizationthat $F$ is dimensionless. We have found them by considering the particularbackground which has one circle and by imposing the corresponding T-dualityconstraint on the independent couplings. In particular, we find that there are12+146 independent gauge invariant couplings at this order, and the T-dualityconstraint can fix 150 of them. We show that these couplings are fullyconsistent with the partial results in the literature. This comparison alsofixes the remaining 8 couplings.

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