Higher-derivative field redefinitions in the presence of boundary

Recently it has been proposed that the consistency with T-duality requires the effective action of string theory at order $\alpha'^n$ satisfies the least action principle provided that the values of the massless fields and their derivatives up to order $n$ are known on the boundary. In this paper we speculate that this boundary condition constrains the field redefinitions and the corrections to the T-duality transformations in the presence of boundary, \eg at order $\alpha'$, the metric does not change, and all other massless fields should change to include only the first derivative of the massless fields. Using the above field redefinitions, we write all gauge invariant bulk and boundary couplings in the bosonic string theory at order $\alpha'$ in a minimal scheme. Then using the assumption that the classical effective action of the string theory at the critical dimension is background independent, we fix the coefficients of the independent gauge invariant couplings by imposing $O(1,1)$ symmetry when the background has a circle and by imposing $O(d,d)$ symmetry when the background has $T^d$. These constraints fix the bulk action up to an overall factor, and the boundary action up to two parameters. By requiring the gravity couplings in the boundary action to be consistent with those in the Chern-Simons gravity, the two boundary parameters are also fixed. Up to the above restricted field redefinitions, the bulk and boundary couplings are exactly those in the K.A. Meissner action and its corresponding boundary action.