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Effective action of string theory at order $α'$ in the presence of boundary

Recently, using the assumption that the string theory effective action at the critical dimension is background independent, the classical on-shell effective action of the bosonic string theory at order $α'$ in a spacetime manifold without boundary has been reproduced, up to an overall parameter, by imposing the $O(1,1)$ symmetry when the background has a circle. In the presence of the boundary, we consider a background which has boundary and a circle such that the unit normal vector of the boundary is independent of the circle. Then the $O(1,1)$ symmetry can fix the bulk action without using the lowest order equation of motion. Moreover, the above constraints and the constraint from the principle of the least action in the presence of boundary can fix the boundary action, up to five boundary parameters. In the least action principle, we assume that not only the values of the massless fields but also the values of their first derivatives are arbitrary on the boundary. We have also observed that the cosmological/one-dimensional reduction of the leading order action in the presence of the Hawking-Gibbons boundary term, produces zero boundary action. Imposing this as another constraint on the boundary couplings at order $α'$, we find the boundary action up to two parameters. For a specific value for these two parameters, the gravity couplings in the boundary become the Chern-Simons gravity plus another term which has the Laplacian of the extrinsic curvature.