  We study the fusion of conformal interfaces in the c=1 conformal field
theory. We uncover an elegant structure reminiscent of that of black holes in
supersymmetric theories. The role of the BPS black holes is played by
topological interfaces, which (a) minimize the entropy function, (b) fix
through an attractor mechanism one or both of the bulk radii, and (c) are
(marginally) stable under splitting. One significant difference is that the
conserved charges are logarithms of natural numbers, rather than vectors in a
charge lattice, as for BPS states. Besides potential applications to
condensed-matter physics and number theory, these results point to the
existence of large solution-generating algebras in string theory.
