Standard Abelian duality transformations in f(T) gravity

According to the perturbation order, the equations of motion of low-energy string effective action are the generalized Einstein equations. Thus, by making use of the conformal transformation of  the metric tensor, it is possible to map the low-energy string effective action into f(T) gravity, relating the dilaton field to the torsion scalar. Considering a homogeneous and isotropic universe and writing the canonical Lagrangian for f(T) gravity, we show that the invariance under the duality transformation holds for the cosmic scale factor a(t)  at the level of the Lagrangian. Finally, by use of  the dualized Lagrangian and also the invariance of torsion scalar under the scale factor duality a(t)→1/a(t),  the specific form of the f(T) function is obtained