segment $[-L,L]$ is mapped to the real positive $w$ axis. Conformal invariance yields $${\cal F}_{\cal C} = -k T \left[ \ln \left(  \frac{\epsilon(L^2-r^2)}{\sqrt{r^4-2 \cos (2 \theta ) r^2 L^2+L^4}}{+}1\right)-\ln 2\right]$$ with $\epsilon=1$ if the object is inside the area encircled by the polymer and $\epsilon=-1$ if it is outside. Computing the force one finds that for $\theta=\pi/2$ the force is radial $f_r=- 2\epsilon k T/[r (1+r^2/L^2)]$ and crosses over from $1/r$ to $L^2/r^3$ as $r$ increases, being attractive if the object is inside, repulsive if it is outside.

Instead of a half-plane one can compute the force in any singly connected domain, as e.g. a disk, or a strip. We consider two distinct infinite strip geometries $z=x+ i y$. In the first, $\cal D$, presented on Fig. \[f:3\], the strip is $0\le y\le L$ and the polymer is attached at $z=0$ and $z= i L$ (in the sense defined above, i.e. passing through a hole at $Z=iL $). Using $w=\tanh(\pi z/(2 L))=(e^{\pi z/L}-1)/(e^{\pi z/L}+1)$ to map it to geometry $\cal A$ of Fig. \[f:1\], one finds the free energy in geometry $\cal D$: $${\cal F}_{\cal D} = - k T \ln \left[ \frac12 + \frac12 \frac{\sqrt 2 \sinh (\pi x/L)}{\sqrt{\cosh(2\pi 