Hagedorn transition and topological entanglement entropy

TL;DR

This paper links the universal entropy reduction in large-N gauge theories on compact manifolds, induced by the Hagedorn transition, to topological entanglement entropy, revealing a topological origin of the entropy deficit.

Contribution

It identifies the universal entropy deficit as topological entanglement entropy and connects bulk and reduced theories, extending the understanding of confinement and deconfinement phases.

Findings

Universal entropy reduction is a topological entanglement entropy.

The topological term arises from Gauss's law constraints.

The results are consistent with holographic calculations for ${\mathcal N}=4$ SYM.

Abstract

Induced by the Hagedorn instability, weakly-coupled $U(N)$ gauge theories on a compact manifold exhibit a confinement/deconfinement phase transition in the large-$N$ limit. Recently we discover that the thermal entropy of a free theory on $\mathbb{S}^3$ gets reduced by a universal constant term, $-N^2/4$, compared to that from completely deconfined colored states. This entropy deficit is due to the persistence of Gauss's law, and actually independent of the shape of the manifold. In this paper we show that this universal term can be identified as the topological entangle entropy both in the corresponding $4+1 D$ bulk theory and the dimensionally reduced theory. First, entanglement entropy in the bulk theory contains the so-called "particle" contribution on the entangling surface, which naturally gives rise to an area-law term. The topological term results from the Gauss's constraint of these surface states. Secondly, the high-temperature limit also defines a dimensionally reduced theory. We calculate the geometric entropy in the reduced theory explicitly, and find that it is given by the same constant term after subtracting the leading term of ${\mathcal O}(\beta^{-1})$. The two procedures are then applied to the confining phase, by extending the temperature to the complex plane. Generalizing the recently proposed $2D$ modular description to an arbitrary matter content, we show the leading local term is missing and no topological term could be definitely isolated. For the special case of ${\mathcal N}=4$ super Yang-Mills theory, the results obtained here are compared with that at strong coupling from the holographic derivation.