  We investigate rotational properties of a system of bosons with attractive
interactions confined in a one-dimensional torus. Two kinds of ground states,
uniform-density and bright-soliton states, are obtained analytically as
functions of the strength of interaction and of the rotational frequency of the
torus. The quantization of circulation appears in the uniform-density state,
but disappears upon formation of the soliton. By comparison with the results of
exact diagonalization of the many-body Hamiltonian, we show that the Bogoliubov
theory is valid at absolute zero over a wide range of parameters. At finite
temperature we employ the exact diagonalization method to examine how thermal
fluctuations smear the plateaus of the quantized circulation. Finally, by
rotating the system with an axisymmetry-breaking potential, we clarify the
process in which the quantized circulation becomes thermodynamically
stabilized.
