Statistical Entropy of Near Extremal Five-Branes

The Hawking Beckenstein entropy of near extremal fivebranes is calculated in terms of a gas of strings living on the fivebrane. These fivebranes can also be viewed as near extremal black holes in five dimensions. 1 e-mail: [email protected] Recently the Bekenstein-Hawking entropy of certain extremal BPS black holes was precisely calculated by counting microstates in string theory [1-4]. Since black holes are non-perturbative objects, the calculations required considerable understanding of nonperturbative string theory and a certain class of solitons called D-branes [5]. These results were extended to leading order away from extremality in [6-9]. These calculations are not so well justified as the BPS ones because one naively expects strong coupling effects in the semiclassical regime in which the black hole picture is valid. Nevertheless it was argued that strong coupling effects could be avoided in certain corners of the parameter space sufficiently near to extremality. The surprising agreement discovered in [6-9] between the string and black hole calculations indicate that under favorable circumstances strong coupling effects could be avoided. The calculation in this paper suffers a similar kind of strong coupling problem, but it extends the ones in [6-9] to a wider class of black holes, finding an interesting interpretation of the classical formulas. In [10,11] the entropy of some near extremal black p-branes was observed to scale as that of a gas of particles living on the brane. The ten dimensional fivebrane that we consider here is a different case which did not fit in the analysis of [11]. In [12] the entropy for BPS fivebranes was calculated in terms of strings living on the fivebrane, here we try to give the description for non-BPS configurations. 1. Classical solutions In ref. [13] general non extremal black hole solutions to type II supergravity compactified on T 5 = T 4 × S1 were considered. These black holes are characterized by three independent charges, their mass and two sizes of the compact space, the radius R of S1 and the volume (2π)V of T . The three charges correspond to the number Q5 of D-fivebranes wrapped on T , the number Q1 of D-strings wrapped on S1 and the momentum N/R flowing along S1. We use the notations of [13]. We will be considering the limit V, γ large and r0 small with V r 2 0 fixed and r 2 0e 2γ fixed. In this limit the black hole is more properly interpreted as a black fivebrane in ten dimensions if the radius R is also large or as a black four-brane in nine extended dimensions if the radius R is small. Alternatively we could think that it is a near extremal black hole in five dimensions with only one charge, the other two charges would be zero or small, of the order of the deviation from extremality. U-duality [14] maps this solution into the solutions describing the near extremal limit of black holes coming from D-branes wrapped on T , or solitonic fivebranes wrapped on