Entanglement Entropy and Cauchy-Hadamard Renormalization
Benoit Estienne, Jiasheng Lin
TL;DR
This work builds a rigorous bridge between 2D conformal field theory and the geometry of surfaces with conical singularities. By defining a renormalized Polyakov anomaly for generalized metrics and studying partition functions on branched covers, it shows that ratios of these partition functions transform under base-metric conformal changes exactly as CFT correlation functions with specific weights $\\Delta_j$. The key result expresses the ramified-cover renormalized relation as $\\frac{\\mathcal{Z}(\\Sigma_d,f^*e^{2h}g)}{\\mathcal{Z}(\\Sigma,e^{2h}g)^d}=e^{-\\sum_j h(w_j)\\Delta_j}\\frac{\\mathcal{Z}(\\Sigma_d,f^*g)}{\\mathcal{Z}(\\Sigma,g)^d}$ with $\\Delta_j=\\frac{c}{12}\\sum_{z\in f^{-1}(w_j)}(\\mathrm{ord}_f(z)-\\frac{1}{\\mathrm{ord}_f(z)})$. The approach relies on a geometric, Green–Stokes-based renormalization that cleanly isolates log-divergent contributions at cone points and demonstrates consistency across reference metrics. This provides a mathematically rigorous framework that underpins the holographic-like interpretation of twist-field correlators and clarifies the replica-trick connections to entanglement entropy in 2D CFTs. The results also clarify the role of conical singularities in spectral geometry and offer precise scaling dimensions tied to ramification data, with connections to existing literature on determinants and twist-field constructions.
Abstract
This note presents a purely geometric construction of the so-called twist-field correlation functions in Conformal Field Theory (CFT), derived from conical singularities. This approach provides a purely mathematical interpretation of the seminal results in physics by Cardy and Calabrese on the entanglement entropy of quantum systems. Specifically, we begin by defining CFT partition functions on surfaces with conical singularities, using a ``Cauchy-Hadamard renormalization'' of the Polyakov anomaly integral. Next, we demonstrate that for a branched cover $f:Σ_d\to Σ$ with $d$ sheets, where the cover inherits the pullback of a smooth metric from the base, a specific ratio of partition functions on the cover to the base transforms under conformal changes of the base metric in the same way as a correlation function of CFT primary fields with specific conformal weights. We also provide a discussion of the physical background and motivation for entanglement entropy, focusing on path integrals and the replica trick, which serves as an introduction to these ideas for a mathematical audience.