String Thermodynamics in D-Brane Backgrounds

TL;DR

The paper analyzes the thermodynamics of weakly coupled strings in D-brane backgrounds near the Hagedorn temperature, addressing whether the Hagedorn temperature is limiting or non-limiting in the canonical ensemble and how microcanonical analysis modifies these conclusions. By computing open- and closed-string densities of states with both canonical and microcanonical methods—including a complex-temperature formalism and a random-walk interpretation—it shows that winding modes in finite-volume D-brane setups can convert non-limiting canonical behaviour into limiting microcanonical behaviour, with energy preferentially flowing to open strings on the largest D-branes. It then develops a qualitative phase diagram for the interacting theory via holographic and black-hole/string correspondences, predicting black-hole–dominated phases bound the Hagedorn regime and highlighting potential cosmological implications such as brane survival and brane–antibrane nucleation. Together, these results illuminate the rich thermodynamics of stringy systems in brane backgrounds and motivate further study of phase structure, finite-volume effects, and cosmological applications in string theory.

Abstract

We discuss the thermal properties of string gases propagating in various D-brane backgrounds in the weak-coupling limit, and at temperatures close to the Hagedorn temperature. We determine, in the canonical ensemble, whether the Hagedorn temperature is limiting or non-limiting. This depends on the dimensionality of the D-brane, and the size of the compact dimensions. We find that in many cases the non-limiting behaviour manifest in the canonical ensemble is modified to a limiting behaviour in the microcanonical ensemble and show that, when there are different systems in thermal contact, the energy flows into open strings on the `limiting' D-branes of largest dimensionality. Such energy densities may eventually exceed the D-brane intrinsic tension. We discuss possible implications of this for the survival of Dp-branes with large values of p in an early cosmological Hagedorn regime. We also discuss the general phase diagram of the interacting theory, as implied by the holographic and black-hole/string correspondence principles.

Figures (4)

• Figure 1: Contour deformation for isolated singularities.
• Figure 2: Variation of $\beta-\beta_c$ with $x=\sqrt{f \, (E-\rho_{c} \,V_{\parallel})}$ for open strings.
• Figure 3: Bulk phase diagram. Only the region $g_s <1$ is represented. The triple point separating the supergravity gas, black hole, and Hagedorn-dominated regimes is located at $g_s \sim 1/\sqrt{V}$, and $E\sim V$. The rightmost region is excluded by the holographic bound (\ref{['holbulk']}).
• Figure 4: World-Volume phase diagram for $g_s <1$. Thick lines represent semiclassical phase transitions or correspondence curves with a major change in the degrees of freedom, whereas dashed lines represent smooth cross-overs within the same basic description. The triple point at low energies was studied in us; it lies at $g_s \sim R_{\parallel}^{p-3}/ N$, $E\sim N^2 / R_{\parallel}$, and is due to finite-size effects in the Yang--Mills theory. The triple point at Hagedorn energies is located at $g_s \sim 1/N$ and $E\sim N^2 \,V_{\parallel}$. The dotted line within the Hagedorn region represents the D-brane bare-mass threshold, $E\sim N\,V_{\parallel}/g_s$. Again, the rightmost region is excluded by the holographic bound (\ref{['holwv']}).