Holographic Gauss-Bonnet transport

TL;DR

This work addresses the challenge of computing shear and bulk viscosities in strongly coupled holographic Gauss-Bonnet theories with unequal UV central charges $c\neq a$, within the causality window $$-\frac{1}{2} \le \frac{c-a}{c} \le \frac{1}{2}$$. It extends the Buchel framework to Gauss-Bonnet gravity, providing explicit horizon-based formulas for $\frac{\eta}{s}$ and $\frac{\zeta}{s}$, including the Gauss-Bonnet case $\frac{\eta}{s}|_{GB}=\frac{1}{4\pi}\left(1+\frac{\lambda_{GB}}{3}(V-12)\right)$ and $9\pi\frac{\zeta}{s}|_{GB}=\sum_i z_{i,0}^2$, with $V$ evaluated at the horizon. The authors test the framework in a simple single-scalar model and in delta${\cal L}_2$ theories, comparing with the Eling-Oz (EO) bulk-viscosity formula, and clarify that the EO extension aligns with the full result when the horizon is effectively two-derivative, while the correct entropy prescription (Wald entropy) is essential for consistency in higher-derivative bulk theories. Their practical guidance, including a detailed treatment of horizon data $z_{i,0}$ and the background equations, enables reliable transport predictions for holographic plasmas with $c\neq a$, advancing understanding of strongly coupled gauge theories at finite coupling and beyond two-derivative gravity. The work thus broadens the domain of holographic transport computations and clarifies the scope of EO-type relations in higher-derivative holography.

Abstract

We extend the computational framework of \cite{Buchel:2023fst} to analysis of shear and bulk viscosities in generic strongly coupled holographic Gauss-Bonnet gauge theories. The finite Gauss-Bonnet coupling constant encodes holographic plasma with non-equal central charges $c\ne a$ at the ultraviolet fixed point. In a simple model we discuss transport coefficients within the causality window $-\frac12\le \frac{c-a}{c}\le \frac 12$ of the theory.

Figures (4)

• Figure 1: Left panel: the bulk viscosity to the entropy density ratios for select values of $(c-a)/a$, see \ref{['acsel']}, in the GB holographic model \ref{['vap']}. The dashed red curves represent the corresponding ratios computed from the EO formula \ref{['eobulk']}. Right panel: the shear viscosity to the entropy density ratios in the same model. The red dots represent the GB conformal gauge theory shear viscosity ratios \ref{['cft']}.
• Figure 2: There is an excellent agreement between the bulk viscosities in the GB holographic model \ref{['vap']}: the bulk viscosity $\zeta$ is computed within the framework of Buchel:2023fst, and the bulk viscosity $\zeta^{EO}$ is extracted from the EO formula. Here we take $\frac{a}{c}=\frac{3}{4}$ (the blue curves in fig. \ref{['figure1']}), as an example.
• Figure 3: At order ${\cal O}(\beta)$ there is a perfect agreement between the ratio of the bulk viscosity to the entropy density evaluated using \ref{['bulks2']} (the solid black curves) and the extension of the EO formula \ref{['eobulk']} to order ${\cal O}(\beta)$ (shown in the red dashed curves), in holographic models which are effectively two-derivative at the horizon.
• Figure 4: Left panel: at order ${\cal O}(\beta)$ there is a disagreement between the ratio of the bulk viscosity to the entropy density evaluated using \ref{['bulks2']} (the solid black curve, which is independently verified in Buchel:2023fst, using the sound wave attenuation computation), and the extensions of the EO formula \ref{['eobulk']} to order ${\cal O}(\beta)$, where the boundary gauge theory thermal state entropy density $s$ is identified with the dual black brane Wald entropy $s=s_{Wald}$ (the red dashed curve), or when the boundary gauge theory thermal state entropy density is identified with the dual black brane Bekenstein entropy $s=s_{Bekenstein}$ (the blue dashed curve). Right panel: ${\cal O}(\beta)$ correction to the speed of the sound waves extracted from the sound wave dispersion relation \ref{['sdisp']} is represented by solid black curve; the computation of the same quantity from the thermodynamic relation \ref{['csth']} with $s=s_{Wald}$ is shown in the red dashed curve, and with $s=s_{Bekenstein}$ is shown in the blue dashed curve. This validates the holographic dictionary $s\equiv s_{Wald}$.