On free energy of 2-d black hole in bosonic string theory

Trying to interpret recent matrix model results (hep-th/0101011) we discuss computation of classical free energy of exact dilatonic 2-d black hole fro m the effective action of string theory. The euclidean space-time action ev aluated on the black hole background is divergent due to linear dilaton vac uum contribution, and its finite part depends on a subtraction procedure. T he thermodynamic approach based on subtracting the vacuum contribution for fixed values of temperature and dilaton charge at the "wall" gives (as in t he leading-order black hole case) S = M/T for the entropy and zero value fo r the free energy F. We suggest that in order to establish a correspondence with a non-vanishing matrix model result for F one may need an alternative reparametrization-invariant subtraction procedure using analogy with non-c ritical string theory (i.e. replacing the spatial coordinate by the dilaton field). The subtraction of the dilaton divergence then produces a finite v alue for the free energy. We also propose a microscopic estimate for the en tropy and energy of the black hole based on the contribution of non-singlet states of the matrix model.