  The least action principle indicates that for the open spacetime manifolds,
there are data on the boundary. Recently, it has been proposed that the data
for the effective actions at order $\alpha'$ are the values of the massless
fields and their first derivatives. These data should be respected by the
T-duality transformations at order $\alpha'$. Moreover, the T-duality
transformations should not change the unit vector to the boundary which in
turns implies that the base space metric should be also invariant. Assuming
such restricted T-duality transformations, we show that the transformation of
the circular reduction of the parity-odd part of the effective action of the
heterotic string theory at order $\alpha'$ under the Buscher rules is cancelled
by some total derivative terms and by some restricted T-duality transformations
at order $\alpha'$. Using the Stokes' theorem, we then show that the boundary
terms in the base space corresponding to the total derivative terms are
$exactly$ cancelled by transformation of the circular reduction of the
Gibbons-Hawking boundary term under the above restricted T-duality
transformations. These calculations confirm the above proposal for the data on
the boundary for the effective actions at order $\alpha'$.

High Energy Physics - Theory