  We present analytical perturbative, and numerical solutions of the Einstein
equation which describe a black hole with a nontrivial dilaton field and a
purely topological gauge potential. The gauge potential has zero field strength
and hence no stress-energy, but it does couple to virtual string worldsheets
which wrap around the Euclidean horizon two-sphere, and generate an effective
interaction in the spacetime lagrangian. We use the lagrangian with a
nonstandard potential for a scalar field that reproduces the effect of the
worldsheet instantons. As has been previously pointed out the topological
charge Q of the gauge field can be detected by an Anharonov-Bohm type
experiment using quantum strings; the classical scalar hair of the solutions
here is a classical detection of Q. ADM mass, dilaton charge and Hawking
temperature are calculated and compared with the known cases when appropriate.
We discuss why these solutions do not violate the no-hair theorems and show
that, for sufficiently small interaction coupling, the solutions are stable
under linear time dependent perturbations.
