How the Hilbert space of two-sided black holes factorises

TL;DR

This work resolves a longstanding holographic puzzle by showing that the bulk Hilbert space associated with a two-sided black hole factorises into a left–right tensor product once non-perturbative wormhole effects are included in the gravitational path integral. Using a resolvent method and an overcomplete basis of bulk states created by matter behind the horizon, the authors demonstrate that the trace over bulk states factorises as $ ext{Tr}_{ ext{bulk}}(k_L k_R)= ext{Tr}_{ ext{L}}(k_L) ext{Tr}_{ ext{R}}(k_R)$ for sufficiently large basis size $K$ (specifically $K\ge d^2$), persisting to all non-perturbative orders in $1/G_N$. They connect this factorisation to a projector-like operator structure and show that the left/right observable algebra becomes a Type I factor, with implications for the algebraic classification of gravitational observables and the nature of subregion entanglement. The paper further extends the analysis to BPS black holes, provides a comprehensive classification of contributing geometries, and discusses generalizations to higher dimensions and charged sectors, highlighting the broad significance for non-perturbative gravity and holography.

Abstract

In AdS/CFT, two-sided black holes are described by states in the tensor product of two Hilbert spaces associated with the two asymptotic boundaries of the spacetime. Understanding how such a tensor product arises from the bulk perspective is an important open problem in holography, known as the factorisation puzzle. In this paper, we show how the Hilbert space of bulk states factorises due to non-perturbative contributions of spacetime wormholes: the trace over two-sided states with different particle excitations behind the horizon factorises into a product of traces of the left and right sides. This precisely occurs when such states form a complete basis for the bulk Hilbert space. We prove that the factorisation of the trace persists to all non-perturbative orders in $1/G_N$, consequently providing a possible resolution to the factorisation puzzle from the gravitational path integral. In the language of von Neumann algebras, our results provide strong evidence that the algebra of one-sided observables transitions from a Type II or Type III algebra, depending on whether or not perturbative gravity effects are included, to a Type I factor when including non-perturbative corrections in the bulk.

Figures (3)

• Figure 1: The contour choice for $\text{Tr}_{\mathcal{H}_{\text{bulk}}(K)}$. The resolvent is defined as a convergent sum in \ref{['eq:resolventdef']} for $\lambda\rightarrow\infty$, so we start from the contour $C_0$ counterclockwise at very large $\lambda$. The resolvent has a branch cut which we show in pink. For $n$ a non-negative integer, we can deform the contour in \ref{['eq:pretraceHbulk']} from $C_0$ to $C_1$ to just include the cut. This requires no pole at $\lambda=0$, which is easy to see for $n>0$. The $n=0$ case is a bit peculiar in the sense that $\mathbf{R}_{ij}(\lambda)$ itself has a pole at $\lambda=0$ but the average $\overline{\mathbf{R}_{ij} \langle q_i|\cdots |q_j\rangle}$ does not. Then we can keep the $C_1$ contour and analytically continue to $n\rightarrow-1$ and deform the contour again from $C_1$ to $C_2$. We then finally arrive at a contour just encircling $\lambda=0$ but clockwise in \ref{['eq:leadingTrHbulk']}.
• Figure 2: The "Page curve" of Hilbert space factorisation. The two-sided partition function as a function of the number of states $K$ used to construct the Hilbert space $\mathcal{H}_\text{bulk}$. For small $K$, we solve the Schwinger-Dyson equation numerically and the resulting two-sided partition function is shown by the blue curve. As $K$ approach $d^2$ we compare the numerical results to the linear perturbative results in $K-d^2$ from \ref{['eq:tr-Hbulk-K-d^2-pert']} shown by the dashed orange line. When $K>d^2$ we confirm that the bulk trace becomes equal to the product of two independent partition functions.
• Figure 3: The value of the differential $d(\beta_L, \beta_R)$ as a function of the number of states $K$ used to construct the Hilbert space $\mathcal{H}_\text{bulk}$ for different values of $\beta_L$ and $\beta_R$. $d(\beta_L, \beta_R)$ increases for small $K$ since the number of states included in the trace and its corresponding value increases. Eventually, as $K$ increases, the bulk partition function is closer to factorising, and that overwhelms the increase in the number of states, making $d(\beta_L, \beta_R)$ decrease. For $K\geq d^2$, we confirm that the bulk trace function factorises by seeing that $d(\beta_L, \beta_R) = 0$ regardless of the value of $\beta_L$ and $\beta_R$.