Holographic stress tensor correlators on higher genus Riemann surfaces

In this work, we present a comprehensive study of holographic stress tensor correlators on general Riemann surfaces, extending beyond the previously well-studied torus cases to explore higher genus conformal field theories (CFTs) within the framework of the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We develop a methodological approach to compute holographic stress tensor correlators, employing the Schottky uniformization technique to address the handlebody solutions for higher genus Riemann surfaces. Through rigorous calculations, we derive four-point stress tensor correlators, alongside recurrence relations for higher-point correlators, within the $\mathrm{AdS}_3/\mathrm{CFT}_2$ context. Additionally, our research delves into the holography of cutoff $\mathrm{AdS}_3$ spaces, offering novel insights into the lower-point correlators of the $T\bar{T}$-deformed theories on higher genus Riemann surfaces up to the first deformation order.