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We show that the fact that the string theory is unoriented does notnecessarily force the Kalb-Ramond field to vanish. We investigated the theoryof the open string propagating in the weakly curved background. The effectiveKalb-Ramond field $B^{eff}_{\mu\nu}$, the background field of the effectivetheory obtained on the solution of the boundary conditions, does not depend onthe $\Omega$-even effective coordinate $q$, but on its T-dual $\tilde{q}$ whichis $\Omega$-odd. This brakes the standard proof that the term with$B_{\mu\nu}^{eff}$ should vanish. From the world-sheet equations of motion weidentify $B_{\mu\nu}^{eff}$ with the torsion potential.

T-dual-coordinate dependence makes the effective Kalb-Ramond field nontrivial