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

TL;DR

This paper investigates the classical free energy of 2D dilatonic black holes in string theory, addressing divergence issues and proposing a subtraction method aligned with matrix model results, along with a microscopic entropy estimate.

Contribution

It introduces an alternative subtraction procedure for free energy calculation and links black hole thermodynamics with matrix model non-singlet states.

Findings

Finite free energy can be obtained with a dilaton-based subtraction method.

Thermodynamic entropy aligns with M/T, consistent with prior results.

Microscopic estimate suggests non-singlet states contribute to black hole entropy.

Abstract

Trying to interpret recent matrix model results (hep-th/0101011) we discuss computation of classical free energy of exact dilatonic 2-d black hole from the effective action of string theory. The euclidean space-time action evaluated on the black hole background is divergent due to linear dilaton vacuum contribution, and its finite part depends on a subtraction procedure. The thermodynamic approach based on subtracting the vacuum contribution for fixed values of temperature and dilaton charge at the "wall" gives (as in the leading-order black hole case) S= M/T for the entropy and zero value for 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-critical string theory (i.e. replacing the spatial coordinate by the dilaton field). The subtraction of the dilaton divergence then produces a finite value for the free energy. We also propose a microscopic estimate for the entropy and energy of the black hole based on the contribution of non-singlet states of the matrix model.