Confinement and chiral symmetry breaking in holography: a smooth switch-off

TL;DR

The paper studies confinement and chiral symmetry breaking in holographic ${\cal N}=4$ SYM compactified on a spatial circle by exploiting an unstable bulk saddle that completes the swallow-tail structure of the thermal free energy. A one-parameter interpolating geometry connects the confining soliton and deconfined black hole phases, allowing confinement (via Wilson loops) and chiral symmetry breaking (via D5-brane embeddings) to be tracked continuously along the unstable branch. The results show that both phenomena persist along the unstable path and fade only at the deconfined endpoint, with the string tension and the quark condensate vanishing as the horizon is approached; the behavior is tightly linked to the boundary stress-energy tensor, suggesting a monopole-density interpretation for confinement. The framework provides a controlled, top-down holographic setting to explore confinement and chiral symmetry breaking and suggests avenues for incorporating finite density or chemical potentials in future work.

Abstract

We revisit the holographic description of the thermal first order phase transition of N=4 SYM compactified on a spatial circle. At the transition, the dominant bulk saddle exchanges between a geometry with a compact spatial circle and one with a compact Euclidean time circle. We construct a one-parameter family of Euclidean geometries that describes the unstable branch of the transition, completing the swallow-tail structure of the free energy. Although these configurations are thermodynamically unstable, they provide a continuous interpolation between the confining soliton and the deconfined black hole phases. Using probe fundamental strings, we show that the theory remains confining along the unstable branch, with a string tension that decreases smoothly and vanishes only in the black hole limit. Introducing fundamental matter via probe D5-branes, we find that chiral symmetry breaking follows the same pattern: the condensate decreases continuously and switches off precisely where confinement disappears. We discuss the implications for the confinement and chiral symmetry breaking mechanisms at large Nc.

Figures (9)

• Figure 1: We show the $\tau-z$ torus on the left as a repeating periodic rectangle. The blue and green lines show identification in the soliton and BH geometries. One of the identifications is always enforced by the metric. The black circles are therefore all the same point. The red lines show the explicitly enforced periodicity of the interpolating metric.
• Figure 2: The free energy of the metrics \ref{['eq:solitonmetric']}, \ref{['eq:BHmetric']}\ref{['eq:intermetric']} as a function of the temperature for $R_z=1/2\pi$, leading to the well-known first-order phase transition at $T=\frac{1}{2\pi R_z}$.
• Figure 3: Numerical solutions for Wilson loops in the geometries with varying $\theta$. We plot the quark-antiquark separation $d$ against the minimum radial depth the U-shaped configurations dip to (in units of $r_0$). The limits $\theta\rightarrow 0$ are the black hole and $\theta\rightarrow \pi/2$ the soliton and for each we recover the known results.
• Figure 4: The $g_{\tau'\tau'}$ component of the induced metric. Only in the strict BH limit ($\theta=0$) does the induced metric develop a worldvolume horizon.
• Figure 5: The D5 brane embeddings in the black hole background. All black lines end in the black hole. The blue lines go to $\rho=0$ and never go inside the black hole. There is a first order phase transition where the solutions overlap with a critical mass at $m_{Crit}/u_0\simeq1.155$ - the red lines are the embeddings that approach the critical mass.