  Double sigma model with the strong constraints is equivalent to the normal
sigma model by imposing the self-duality relation. The gauge symmetries are the
diffeomorphism and one-form gauge transformation with the strong constraints.
We modify the Dirichlet and Neumann boundary conditions with the fully $O(D,
D)$ description from the doubled gauge fields. We perform the one-loop $\beta$
function for the constant background fields to find low energy effective theory
without using the strong constraints. The low energy theory can also be
$O(D,D)$ invariant as the double sigma model. We use the other one way to
construct different boundary conditions from the projectors. Finally, we
combine the antisymmetric background field with the field strength to redefine
a different $O(D, D)$ generalized metric. We use this generalized metric to
construct a consistent double sigma model with the classical and quantum
equivalence. We show the one-loop $\beta$ function for the constant background
fields and obtain the normal sigma model after integrating out the dual
coordinates.
