It has been recently observed that the imposition of the $O(1,1)$ symmetry on the circle reduction of the classical effective action of string theory, can fix the effective action of the bosonic string theory at order $\\\\\\\\alpha\\\\\\\'^2$, up to an overall factor. In this paper, we use the cosmological reduction on the action and show that, up to one-dimensional field redefinitions and total derivative terms, it can be written in the $O(25,25)$-invariant form proposed by Hohm and Zwiebach.

Keywords