Generator of conformal transformation in presence of conformal defect

Question

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 $x=0$ ?

Answer 1

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