Topological aspects of generalized gravitational entropy

TL;DR

This work analyzes whether the LM replica construction in holography inherently enforces the homology constraint on the RT/HRT extremal surface. It shows that the homology condition is equivalent to a topological consistency requirement: the bulk geometry with a conical defect along ${\mathcal E}$ must lift to a smooth $q$-fold branched cover for every positive integer $q$. Through explicit counterexamples (e.g., a torus with a crosscap) and a rigorous cohomological/topological proof, the authors prove that lifting for all $q$ is necessary and sufficient for homology, thereby linking local LM manipulations to the global RT prescription. The results rely on relative cohomology, Poincaré–Lefschetz duality, and Thom class considerations, and extend to non-orientable manifolds via twisted coefficients. Overall, the paper clarifies the precise topological prerequisites for holographic entanglement entropy calculations and underscores the interplay between replica symmetry and global spacetime topology.

Abstract

The holographic prescription for computing entanglement entropy requires that the bulk extremal surface, whose area encodes the amount of entanglement, satisfies a homology constraint. Usually this is stated as the requirement of a (spacelike) interpolating surface that connects the region of interest and the extremal surface. We investigate to what extent this constraint is upheld by the generalized gravitational entropy argument, which relies on constructing replica symmetric q-fold covering spaces of the bulk, branched at the extremal surface. We prove (at the level of topology) that the putative extremal surface satisfies the homology constraint if and only if the corresponding branched cover can be constructed for every positive integer q. We give simple examples to show that homology can be violated if the cover exists for some values of q but not others, along with some other issues.

Figures (3)

• Figure 1: Replica construction in the boundary and bulk for $q=3$. The replica symmetric $q$ copies of the field theory on $\mathcal{B}$, form a $q$-fold branched cover $\tilde{\mathcal{B}}_q$ which fixes the asymptotic data for the bulk problem. The bulk covering spacetime $\tilde{\mathcal{M}}_q$ has a ${\mathbb Z}_q$ symmetry with fixed point locus $\mathbf{e}_q$ (shown as the wavy lines) anchored on $\partial \mathcal{A}$. Typically one also encounters via this construction a bulk interpolating surface $\mathbf{r}_q$ (the light blue branching surface) in the bulk whose boundaries are $\mathbf{e}_q$ and $\mathcal{A}$. Conventional intuition dictates that the bulk spacetimes are all covers over a single fundamental domain (one of the components in the picture) branched over the codimension-1 surface $\mathbf{r}_q$. Passing through this surface cycles through the sheets of the bulk in a fashion identical to passage through $\mathcal{A}$. The homology condition posits that such an $\mathbf{r}_q$ exists. We argue that this picture is accurate as long as we are suitably careful with the notion of allowed branched covers. As $q \rightarrow 1$, $\mathbf{r}_q \rightarrow \mathcal{R}_\mathcal{A}$ and $\mathbf{e}_q \rightarrow \mathcal{E}$.
• Figure 2: Different ways of filling the boundary torus. The replica $\mathbb{Z}_q$ symmetry acts as a rotation along the Euclidean time direction by $\beta$. Case (a) shows a slice of $\tilde{\mathcal{M}}_3$ after filling the boundary time circle with a disk; the replica fixed point set $\mathbf{e}_3$ (blue) is a codimension-2 surface in the centre of the circle. Cases (b), (c) and (d) show slices of $\tilde{\mathcal{M}}_q$ for $q=2,3,4$, respectively, after filling the boundary torus with a cross cap. For $q=2$ the fixed point set $\mathbf{e}_2$ is a codimension-1 orbifold plane wrapping the cross cap. For $q\geq 3$ there is no fixed point set under $\mathbb{Z}_q$ (i.e. $\mathbf{e}_q$ is empty) and the homology constraint is violated. However, we indicate in (d) that for even $q$ the cross cap itself is still a fixed point set under the subgroup $\mathbb{Z}_2$ of rotations by $\frac{q}{2}\beta$, resulting in an orbifold plane under the quotient.
• Figure 3: Three different candidates for bulk geodesics (blue) whose length may be considered to compute the entanglement entropy of the spatial boundary interval ${\mathcal{A}}$ (red). The green surfaces $\mathcal{D}$ illustrate the connection between boundary sheet counting (intersections with $\mathcal{A}$) and bulk intersections. Thick dots illustrate intersections. The bulk surfaces $\mathcal{E}^{(a)}$ and $\mathcal{E}^{(c)}$ are homologous to $\mathcal{A}$; which one of them is dominant depends on the length of $\mathcal{A}$. The surface $\mathcal{E}^{(b)}$ is not homologous to ${\mathcal{A}}$ and is also forbidden by our topological consistency condition. In this latter case, boundary intersections do not match bulk intersection numbers.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof