invariance means that ${\cal P}(z,\bar z)={\cal P}_0(g(z),\overline{g(z)})$ where ${\cal P}_0$ is the upper-half plane result given above. We find for the free energy and force $$\begin{aligned}
 {\cal F}_{\cal B} &=& - kT\left[ \ln ( 1 + \cos(\alpha \theta)) - \ln2\right] \\
 \vec f_{\cal B}&=&- k T \frac{\vec n_\theta}{r} \frac{\pi}{\phi} \tan(\pi \theta/2 \phi)\ .\end{aligned}$$ Let us now study a polymer with [*two endpoints fixed*]{} as shown in Fig. \[f:2\] (geometry ${\cal C}$). Note that since SLE describes the continuum limit of the SAW with fixed endpoints but fluctuating number of steps $N$ at the critical chemical potential [@SchrammSAW; @Kennedy], a possible setting for an experiment is to consider the real axis as impenetrable, fix one endpoint at $x=-L$ and place a hole at $x=L$, through which the self-avoiding polymer passes. Note that it is also possible to use two symmetric holes. Assuming equilibrium for an infinitely long polymer ensures that the chemical potential is at its critical value. We can now use $w=g(z)= \frac{z+L}{L-z}$ which maps geometry $\cal C$ back to $\cal A$. It maps the half plane onto itself, preserves the real axis, and maps $z=-L$ to $w=0$ and $z=L$ to infinity, hence back to Fig. 1. Note that the 