We discuss some aspects related to holography of anti–de Sitter (AdS) dyonic hairy black holes, which break the conformal symmetry of the boundary. We use counterterms for the scalar field that satisfies mixed boundary conditions to compute the Euclidean action and dual stress tensor. We apply these results to show that the Cardy-Verlinde formula is not satisfied. However, when the magnetic (or electric) charge vanishes, the conformal symmetry is preserved and the entropy of the black hole can be put in the Cardy-Verlinde form. In our analysis, there is no need of adding extra finite counterterms and, in this particular case, we explicitly show that our results match the ones obtained from using the superpotential as a counterterm.

中文翻译：

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AdS 毛状黑洞的全息图和 Cardy-Verlinde 公式

我们讨论了与反德西特 (AdS) 动力毛状黑洞全息相关的一些方面，这些方面破坏了边界的共形对称性。我们使用满足混合边界条件的标量场的反项来计算欧几里得作用和对偶应力张量。我们应用这些结果表明不满足 Cardy-Verlinde 公式。然而，当磁（或电）电荷消失时，共形对称性得以保留，黑洞的熵可以采用 Cardy-Verlinde 形式。在我们的分析中，不需要添加额外的有限反项，在这种特殊情况下，我们明确表明我们的结果与使用超势作为反项获得的结果相匹配。