T-duality constraint on effective Lagrangians

Recent studies have highlighted the significant role of utilizing O(1, 1) symmetry in the circular reduction of effective actions to determine NS-NS couplings in the effective action of string theory. However, these calculations often result in residual terms as total derivatives that do not conform to O(1, 1) transformations. In this paper, we present explicit calculations at alpha '\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha '$$\end{document} order, demonstrating the enforceability of this symmetry on effective Lagrangians to establish the parameters governing covariant couplings in any scheme. Notably, we discover the O(1, 1)-invariant Lagrangians corresponding to the Metsaev-Tseytlin action and the Meissner action.