Twisting the Hagedorn temperature in planar $\mathcal{N}=4$ super Yang-Mills

TL;DR

This work extends the quantum spectral curve framework to compute the Hagedorn temperature $T_H$ of planar $ ext{N}=4$ SYM at finite temperature with twists for both $R$-charges and spins. By combining weak-coupling bead counting, strong-coupling string theory, and twisted QSC numerics, it derives analytic weak-coupling results (including one-loop for arbitrary μ and two-loop at μ=1/2) and explicit strong-coupling expansions, while numerically validating the results and testing Harmark's covariant zero-point shift. The study shows that the QSC accurately connects weak- and strong-coupling regimes, providing evidence for the proposed worldsheet corrections and offering detailed predictions for twisted thermodynamics. These findings deepen understanding of Hagedorn physics in AdS/CFT and demonstrate the robustness of the twisted QSC approach for finite-temperature observables with chemical potentials.

Abstract

We consider planar $\mathcal{N}=4$ super Yang-Mills at finite temperature with chemical potentials that couple either to the $R$-charges or the spins of the operators. We find expressions for the Hagedorn temperatures at both zero coupling by explicitly counting states, and at strong coupling using the string theory dual. We then apply the quantum spectral curve (QSC) to this problem, which adds additional twists to the $Q$-functions. For a single chemical potential $μ$ coupled to one of the $R$-charges, we find the analytic weak-coupling Hagedorn temperature to one-loop order for any value of $μ$, and to two-loop order for $μ=1/2$. We then solve the QSC numerically, showing that at strong coupling there is good agreement with the string theory prediction to order $1/λ^{1/4}$. This provides further evidence for a recent conjecture of Harmark for the form of the world-sheet zero-point shift. We also use the QSC to find the analytic one-loop correction to the Hagedorn temperature with non-zero chemical potentials coupled to the spins.

Figures (11)

• Figure 1: The Hagedorn temperature at zero coupling as a function of the chemical potential at zero coupling Harmark:2006di.
• Figure 2: We display the 1-loop correction to the Hagedorn temperature for $\mu\in [0,1]$. The solid blue line is the exact analytic expression \ref{['eq:TH1Analytic']} while the red points are from fitting the numerical QSC data shown in \ref{['fig:weakCoupling']}.
• Figure 3: The Hagedorn temperature at zero coupling in ${\cal N}=4$ SYM against both chemical potentials for the spin.
• Figure 4: In pink we show the numerical solution of the QSC in the range $g^2\leq0.1$. This is a total of 950 data points. In blue we show again the zero coupling result. In green we show the zero twist results of Harmark:2021qma.
• Figure 5: The strong coupling data (1645 points) is shown in blue. In pink we show the weak coupling data from \ref{['fig:weakCoupling']} and in green we show the zero twist data from Ekhammar:2023glu.