Black holes and five-brane thermodynamics

The phase diagram for $\mathrm{D}p$-branes in M theory compactified on ${\mathrm{T}}^4,{\mathrm{T}}^4{/Z}_2,{\mathrm{T}}^5,$ and ${\mathrm{T}}^6$ is constructed. As for the lower-dimensional tori considered in our previous work [E. Martinec and V. Sahakian, Phys. Rev. D 59, 124005 (1999)], the black brane phase at high entropy connects onto matrix theory at low entropy; we thus recover all known instances of matrix theory as consequences of the Maldacena conjecture. The difficulties that arise for ${\mathrm{T}}^6$ are reviewed. We also analyze the D1-D5 system on ${\mathrm{T}}^5;$ we discuss its relation to matrix models of M5-branes, and use spectral flow as a tool to investigate the dependence of the phase structure on angular momentum.