Black Holes and Five-brane Thermodynamics

TL;DR

This work extends Maldacena-based thermodynamic phase analyses to Dp-branes on various tori, revealing that high-entropy black-brane phases universally map to matrix-theory-like descriptions and that the low-entropy regimes relate to LC/M-theory geometries via U-duality, with p=6 presenting a notable departure due to gravity not decoupling. It provides a detailed, patch-by-patch construction of phase diagrams for p=4,5,6 and for the D1-D5 system on T^4×S^1 and T^4/Z_2×S^1, uncovering a rich web of dualities, localization transitions, and Horowitz-Polchinski correspondence points. A central technical component is the use of spectral flow in the 1+1d N=(4,4) superconformal algebra, which links NS and R sectors and constrains the density of states via a Cardy-like formula, thereby illuminating how angular momentum reorganizes phase structure. The results sharpen the dictionary between gravitational black-hole thermodynamics and non-gravitational dual descriptions (Matrix strings, little-string theories, and heterotic/M-theory matrix models), with implications for DLCQ descriptions of M5-branes and the emergence of matrix-string phases in diverse compactifications.

Abstract

The phase diagram for Dp-branes in M-theory compactified on $T^4$, $T^4/Z_2$, $T^5$, and $T^6$ is constructed. As for the lower-dimensional tori considered in our previous work (hep-th/9810224), 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 $T^6$ are reviewed. We also analyze the D1-D5 system on $T^5$; we exhibit 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.

Figures (6)

• Figure 1: Phase diagram of the six-dimensional $(2,0)$ theory on $T^4\times S^1$. $S$ is entropy, $V=R/{\it l}_{{\rm pl}}$ is the size of a cycle on the $T^4$ of light-cone M theory, and $N$ is longitudinal momentum quantum. The dotted lines denote symmetry transformations: M: M lift or reduction; T: T duality; S: S duality. The solid lines are phase transition curves. Double solid lines denote correspondence curves. The dashed line is the extension of the axis $V=1$, and is merely included to help guide the eye. The label dictionary is as follows: $D0$: black D0 geometry; $W11$: black 11D wave geometry; $11DBH$: 11D LC black hole; $\overline{D0}$: black smeared D0 geometry; $\overline{W11}$: black smeared 11D wave geometry; $\overline{11D}BH$: 11D smeared LC black hole; $D4$: black D4 geometry; $M5$: black M5 geometry; $\overline{F1}$: black smeared fundamental string geometry; $\overline{WB}$: black smeared IIB wave geometry; $\overline{10D}BH$: IIB boosted black hole. The phase diagram can also be considered that of the $(2,0)$ theory on $T^4/Z_2\times S^1$ by reinterpreting the $\overline{F1}$, $\overline{WB}$, $\overline{10D}$ phases, and the Matrix string phase as those of a Heterotic theory. .
• Figure 2: Phase diagram of 'little string' theory on $T^5$. The labeling is as in Figure \ref{['SYM5fig']}. $D0$: black D0 geometry; $W11$: black 11D wave geometry; $11DBH$: 11D LC black hole; $\overline{D0}$: black smeared D0 geometry; $\overline{W11}$: black smeared 11D wave geometry; $\overline{11D}BH$: 11D smeared LC black hole; $D5$: black D5 geometry; $NS5B$: black five branes in IIB theory; $NS5A$: black five branes in IIA theory; $M5$: black M5 brane geometry; $\widetilde{M5}$: black smeared M5 brane geometry; $\widehat{M}\overline{W11}$: black smeared wave geometry in $\widehat{M}$ theory; $\widehat{M}W11$: black smeared wave geometry in the $\widehat{M}$ theory; $\widehat{11D}BH$: smeared boosted black holes in the $\widehat{M}$ theory.
• Figure 3: Phase diagram of the $D6$ system. $S$ is entropy, $V=R/{\it l}_{{\rm pl}}$ is the size of a cycle on the $T^6$ of the LC M theory, and $N$ is longitudinal momentum. The dotted lines are symmetry transformations: M: M lift or reduction; T: T duality; S: S duality. The solid lines are phase transition curves. Double solid lines denote correspondence curves. The label dictionary is as follows: $\overline{M}TN$,$\widehat{M}TN$: black Taub-NUT geometry; $D6$,$\widetilde{D6}$: black D6 geometry; $D0$,$\widetilde{D0}$: black D0 geometry; $W11$,$\widetilde{W11}$: black 11D wave geometry; $11DBH$,$\widetilde{11D}BH$: 11D LC black hole; $\overline{D0}$,$\widetilde{\overline{D0}}$: black smeared D0 geometry; $\overline{W11}$,$\widetilde{\overline{W11}}$: black smeared 11D wave geometry; $\overline{11D}BH$,$\widetilde{\overline{11D}}BH$: 11D smeared LC black hole.
• Figure 4: Thermodynamic phase diagram of 'little strings' wound on the $S^1$ of $T^4\times S^1$, with $Q_1$ units of winding and $Q_5$ five branes. $k\equiv Q_1 Q_5$ and $1<q\equiv Q_1/Q_5<k$. $g_6$ is the six dimensional string coupling of the $D1D5$ phase. The label dictionary is as follows: $D1D5$: black $D1D5$ geometry; $NS5FB$: black NS5 geometry with fundamental strings in IIB theory; $D0D4$: black D0D4 geometry; $\overline{D0D4}$: black smeared D0D4 geometry; $M5W$: black boosted M5 brane geometry; $\overline{M5W}$: black smeared boosted M5 brane geometry; $NS5WA$: black boosted NS5 branes in IIA theory; $F1WB$: black boosted fundamental strings in IIB theory; $\overline{F1WB}$: black smeared and boosted fundamental strings in IIB theory; L: localization transitions.
• Figure 5: Allowed region for states belonging to unitary representations of the (NS) superconformal algebra. The dashed curve represents the continuous spectral flow $h_\eta=j_\eta^2/k$ of the point $h=j=0$. Spectral flow slides the boundary polygon along the parabola; a half unit of flow gives the Ramond sector (inset).