Nongeometric background arising in the solution of Neumann boundary conditions

We investigate the open string propagation in the weakly curved background with the Kalb-Ramond field containing an infinitesimal part, linear in coordinate. Solving the Neumann boundary conditions, we find the expression for the space-time coordinates in terms of the effective ones. So, the initial theory reduces to the effective one. This effective theory is defined on the nongeometric doubled space $(q^{\mu },{\tilde{q}}_{\mu })$, where $q^{\mu }$ is the effective coordinate and ${\tilde{q}}_{\mu }$ is its T-dual. The effective metric depends on the coordinate $q^{\mu }$ and there exists non-trivial effective Kalb-Ramond field which depends on the T-dual coordinate ${\tilde{q}}_{\mu }$. The fact that ${\tilde{q}}_{\mu }$ is $\mathrm{\Omega }$-odd leads to the nonvanishing effective Kalb-Ramond field.