Holographic R\'enyi $n\to 0$ entropy and Euclidean fluids

We explore the holographic prescription for computing the refined R\'enyi entropies $\tilde S_n$ in the $n \to 0$ limit within the AdS$_{d+1}$/CFT$_d$ framework. This limit can be interpreted as a high-temperature regime with respect to the energy defined by the modular Hamiltonian of the state reduced to a subregion. To leading order in $n$, we find that the system attains local equilibrium and admits a CFT description in terms of a Euclidean, irrotational perfect fluid. This fluid exhibits vortex-like boundary conditions at the entangling surface. Guided by this physical picture, we construct an ansatz for the dual geometry in terms of the boundary fluid variables. We show that our anzats solves Einstein's equations coupled to a cosmic brane at leading order in $n$, in agreement with Dong's proposal for the holographic dual to the refined R\'enyi entropy. The resulting conical singularity, signaling the brane's location, can be understood from this perspective as the bulk extension of the boundary fluid vortices.