from $+i\epsilon$ to $+i\infty$.

Extended Logarithmic world-sheet Algebra of recoil in RW backgrounds
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Following the flat space time analysis of [@kmw], we now proceed to discuss the conformal structure of the recoil operators in RW backgrounds. We shall do so by acting on the operator ${\cal }D^{(q)}$ (\[opd\]) with the world-sheet energy momentum tensor operator $T_{zz} \equiv T$ (in a standard notation). Due to the form of the background space time (\[rwmetric\]), the stress tensor $T$ assumes the form $$\label{stress}  
2T =-(\partial t)^2 + a^2(t) (\partial X^i)^2$$ where, from now on, $\partial \equiv \partial _z$, unless otherwise stated. One can then obtain the relevant operator-product expansions (OPE) of $T$ with the operators ${\cal D}^{(q)}$. For convenience in what follows we shall consider the action of each of the two terms in (\[stress\]) on the operators ${\cal D}^{(q)}$ separately. For the first (time $t$-dependent part), one has, as $z \to w$: $$\begin{aligned}
&~& -\frac{1}{2}(\partial t(z))^2\cdot{\cal D}^{(q)}(w) 
=  \frac{v_i}{(z-w)^2}\left[
N_q\int _{-\infty}^{+\infty}d\omega 
\frac{\omega^2/2}{(\omega -i\epsilon)^{q+1}}~e^{i\omega~t(w)}\right]
\partial_nX^i 
= \nonumber \\
&~& \frac{1}{(z-w)^2}\left[-\frac{\epsilon^2}{2}{\cal D}^{(q)} + q\epsilon {\cal D}^{(q-1)} + \frac{q(q-1)}{2}{\cal D}^{(q-2)} \right]\end{aligned}$$ The above formulæ were derived for asymptotically large time $t$, assuming the two-point correlators $$\label{correl}
\langle X^\mu(z) X^\nu(w) \rangle 
= 2G^{\mu\nu}{\rm ln}|z-w|^2 + 