Boundary conditions and the generalized metric formulation of the double sigma model

Double sigma model with strong constraints is equivalent to the ordinary sigma model by imposing a self-duality relation. The gauge symmetries are the diffeomorphism and one-form gauge transformation with the strong constraints. We consider boundary conditions in the double sigma model from three ways. The first way is to modify the Dirichlet and Neumann boundary conditions with a fully O(D,D)O(D,D) description from double gauge fields. We perform the one-loop β   function for the constant background fields to find low-energy effective theory without using the strong constraints. The low-energy theory can also have O(D,D)O(D,D) invariance as the double sigma model. The second way is to construct different boundary conditions from the projectors. The third way is to combine the antisymmetric background field with field strength to redefine an O(D,D)O(D,D) generalized metric. We use this generalized metric to reconstruct a consistent double sigma model with the classical and quantum equivalence.