Generator of conformal transformation in presence of conformal defect I was reading about conformal defects, when I came across the following:

Consider some free massless scalar field in 1+1 dimension $\phi(x,t)$ living on a world sheet seperated by some defect at $x=0$ and and Energy-Momentum tensor given by
$$T_{xt}  = T_{++}-T_{--}  =\partial_x \phi \partial_t \phi$$
where $x^{\pm} = t \pm x , \partial_\pm = \frac{1}{2}(\partial_t \pm \partial_x)$.
Then the conformal transformations leaving the $x=0$ worldline invariant are generated by the operators
$$(f(x^+)T_{++} - f(x^-)T_{--})$$

I am not sure how to see/prove this. Does generator here mean that the operator given by $exp(-ia(f(x^+)T_{++} - f(x^-)T_{--})$ with $a$ some constant, acting on some function $f(x^\mu)$ does not change it's value at $x=0$ ?
 A: I now had some time to think about this again and I think I came up with an explanation. I would greatly appreciate it if somebody had some comments whether this is reasonable:
As we know from 2-dimensional CFT where we usually like to work with the variables $z = t+ix$ resp. $\overline{z} = t-ix$  hence holomorphic/antiholomorphic expressions we have:
$$T_{++} \leftrightarrow T_{zz} = T(z) \quad , \quad T_{--} \leftrightarrow T_{--} = T(\overline{z})$$
Sine we can express $T(z)$ resp. $T(\overline{z})$ ad Laurent-series as
$$T(z)  = \sum z^{-n-2}L_n $$
where the operators $L_n$  form a Virasoro algebra and generate the conformal transfromations.
Since the physical transformations like dilations/rotations are actually generated by the sum/difference of such generators the statement that $(f(x^+)T_{++} - f(x^-)T_{--})$ generates the conformal transformations of our theory seems reasonable.
Furthermore since at point for which x=0 we have
$$x^+ = t + 0 = t- 0 = x^-$$
and therefore
$$(f(x^+)T_{++} - f(x^-)T_{--}) = f(t)T(t) -f(t)T(t) = 0,$$
hence the above operator leaves invariant the line $x=0$. Where the implicit statement $T_{++} = T(x^+)$ follows from the one for $T_{zz}$ by Wick rotation
