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

TL;DR

This paper investigates how non-commutativity parameters in open string theory depend on both the effective coordinate and its T-dual, especially in a background with constant and linear Kalb-Ramond fields, revealing new coordinate-dependent effects.

Contribution

It extends previous models by showing that in a combined background, the non-commutativity parameter depends on the T-dual of the effective coordinate, introducing additional small, yet significant, terms.

Findings

Effective metric becomes coordinate dependent.

Non-trivial effective Kalb-Ramond field appears.

Non-commutativity parameters depend on T-dual variables.

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.