Thermodynamics of Large-N Super Yang-Mills Theory and AdS/CFT Correspondence

TL;DR

This paper investigates whether the thermodynamic free-energy density $F$ of large-$N$ ${\cal N}=4$ SU($N$) SYM can interpolate smoothly between weak and strong 't Hooft coupling as predicted by AdS/CFT. It first computes $F$ perturbatively up to ${\cal O}((g_{\rm YM}^2 c_A)^{3/2})$, including Debye screening and non-analytic contributions, showing a fixed-order ${\cal O}((g_{\rm YM}^2 c_A)^{3/2})$ term with a sign that initially hinders monotonic interpolation. It then applies Padé approximants to the fixed-order series, constructing two forms $R_{[1,2]}$ and $R_{[2,1]}$, which yield a monotonic decrease of $F$ with increasing coupling, aligning with the strong-coupling AdS$_5$ gravity result and supporting the possibility of smooth interpolation. The authors also discuss the D1–D5 system and an AdS$_3$ black hole case, noting a leading-order agreement of weak and strong coupling free energies in 2D that hints at a potential non-renormalization phenomenon, thus highlighting dimension-specific subtleties that may affect interpolation. Overall, the work advocates a Padé-based approach as a practical tool for probing crossovers in gauge/gravity dualities while outlining limitations and directions for higher-order or nonperturbative analyses.

Abstract

Thermodynamics of d=4, N=4 supersymmetric SU(N) Yang-Mills theory is studied with particular attention on perturbative expansion at weak `t Hooft coupling regime and interpolation to strong coupling regime thereof. Non-ideal gas effect to free-energy is calculated and found that leading- and next-to-leading-order corrections contribute with relative opposite sign. Pade approximant method is adopted to improve fixed-order, perturbative series and is found to decrease free-energy monotonically as `t Hooft coupling parameter is increased. This may be regarded as an indication of smooth interpolation of thermodynamics between weak and strong `t Hooft coupling regimes, as suggested by Maldacena's AdS/CFT correspondence.

Figures (5)

• Figure 1: One-loop Feynman diagrams contributing to free-energy $F_1$. Gauge bosons are represented by wiggly lines, ghosts by dotted lines, fermions by solid lines, and scalars by dashed lines.
• Figure 2: Two-loop diagrams contributing to the free-energy density $F_2$.
• Figure 3: One-loop self-energy diagrams for gauge bosons and scalars.
• Figure 4: Mass-counterterm diagrams due to thermal Debye masses.
• Figure 5: Plots of fixed-order perturbative result, and Padé approximants $R_{[1,2]}$ and $R_{[2,1]}$.