Non-commutativity parameters depend not only on the effective coordinate but on its T-dual as well

Abstract

We extend our investigations of the open string propagation in the weakly curved background to the case when Kalb-Ramond field, beside the infinitesimal term linear in coordinate $B_{\mu\nu\rho}x^\rho$, contains the constant term $b_{\mu\nu}\neq 0$. In two previously investigated cases, for the flat background ($b_{\mu\nu}\neq 0$ and $B_{\mu\nu\rho}=0$) and the weakly curved one ($b_{\mu\nu}= 0$ and $B_{\mu\nu\rho}\neq 0$) the effective metric is constant and the effective Kalb-Ramond field is zero. In the present article ($b_{\mu\nu}\neq 0$ and $B_{\mu\nu\rho}\neq 0$) the effective metric is coordinate dependent and there exists non-trivial effective Kalb-Ramond field. It depends on the $\sigma$-integral of the effective momenta $P_\mu(\sigma)=\int_{0}^{\sigma}d\eta p_\mu(\eta)$, which is in fact T-dual of the effective coordinate, $P_\mu=\kappa g_{\mu\nu}{\tilde{q}}^\nu$. Beside the standard coordinate dependent term $\theta^{\mu\nu}(q)$, in the non-commutativity parameter, which is nontrivial only on the string end-points, there are additional $P_\mu$ (or ${\tilde{q}}^\mu$) dependent terms which are nontrivial both at the string endpoints and at the string interior. The additional terms are infinitesimally small. The part of one of these terms has been obtained in Ref. \cite{HY} and the others are our improvements.