Entanglement and Topology in RG Flows Across Dimensions: Caps, Bridges and Corners

TL;DR

This paper explores how entanglement entropy evolves during RG flows from higher to lower-dimensional CFTs in holographic settings, revealing universal topological and geometric transitions influenced by compactification details.

Contribution

It generalizes entanglement entropy analysis to setups without full Lorentz symmetry and uncovers universal phase transitions linked to the topology of compact spaces.

Findings

Universal transitions from cap to bridge to corner phases in entanglement entropy

Topology and geometry of compact space influence entanglement evolution

Holographic analysis applied to twisted compactifications of 4d ${ m N}=4$ SYM

Abstract

We quantitatively address the following question: for a QFT which is partially compactified, so as to realize an RG flow from a $D$-dimensional CFT in the UV to a $d$-dimensional CFT in the IR, how does the entanglement entropy of a small spherical region probing the UV physics evolve as the size of the region grows to increasingly probe IR physics? This entails a generalization of spherical regions to setups without full Lorentz symmetry, and we study the associated entanglement entropies holographically. We find a tight interplay between the topology and geometry of the compact space and the evolution of the entanglement entropy, with universal transitions from `cap' through `bridge' and `corner' phases, whose features reflect the details of the compact space. As concrete examples we discuss twisted compactifications of 4d ${\cal N}=4$ SYM on $T^2$, $S^2$ and hyperbolic Riemann surfaces.