Higher Spin de Sitter Hilbert Space

TL;DR

<3-5 sentence high-level summary> This work constructs a complete microscopic Hilbert space for minimal higher spin gravity in de Sitter space by introducing 2N boundary bosons Q_x^α, organizing gauge-invariant information into a 2N×2N matrix model, and defining a normalizable Hartle-Hawking vacuum. It shows perturbative bulk expectations are recovered in a large-N limit via shadow-transform relations to Sp(N) data, while providing exact, nonperturbative control for late-time vacuum correlators and probabilities of large field excursions. Bulk reconstruction is possible only in a coarse-grained sense, with an exponentially small error term e^{-cN}, and breaks down beyond a finite spatial resolution set by N, consistent with de Sitter entropy bounds. The physical Hilbert space after gauging the higher-spin symmetry becomes 𝓗_phys, which remains infinite-dimensional but is effectively described by a finite 2N×2N matrix model for gauge-invariant observables, realizing a concrete form of cosmological complementarity.

Abstract

We propose a complete microscopic definition of the Hilbert space of minimal higher spin de Sitter quantum gravity and its Hartle-Hawking vacuum state. The fundamental degrees of freedom are $2N$ bosonic fields living on the future conformal boundary, where $N$ is proportional to the de Sitter horizon entropy. The vacuum state is normalizable. The model agrees in perturbation theory with expectations from a previously proposed dS-CFT description in terms of a fermionic Sp(N) model, but it goes beyond this, both in its conceptual scope and in its computational power. In particular it resolves the apparent pathologies affecting the Sp(N) model, and it provides an exact formula for late time vacuum correlation functions. We illustrate this by computing probabilities for arbitrarily large field excursions, and by giving fully explicit examples of vacuum 3- and 4-point functions. We discuss bulk reconstruction and show the perturbative bulk QFT canonical commutations relations can be reproduced from the fundamental operator algebra, but only up to a minimal error term $\sim e^{-\mathcal{O}(N)}$, and only if the operators are coarse grained in such a way that the number of accessible "pixels" is less than $\mathcal{O}(N)$. Independent of this, we show that upon gauging the higher spin symmetry group, one is left with $2N$ physical degrees of freedom, and that all gauge invariant quantities can be computed by a $2N \times 2N$ matrix model. This suggests a concrete realization of the idea of cosmological complementarity.

Figures (8)

• Figure 2.1: Random cosmological samples of dS$_{3+1}$ higher spin field modes $\beta$ of spin $s=0,2,4,6$ in position space. Drawn from Gaussian distribution with power spectrum ${\cal D}^{-s+\frac{1}{2}} \sim k^{-2s+1}$ where ${\cal D}$ is the discrete Laplacian on a 3d torus of linear size $L$, with lattice spacing $\frac{L}{200}$ (so $K = 200^3 = 8 \times 10^6$ lattice points), with the zero mode IR-regulated by adding a mass term of order $\frac{1}{L}$ to ${\cal D}$.
• Figure 3.1: Wave function squared $|\tilde{\psi}_{\rm HH}(b^0)|^2$ for $N=2$ according to naive interpretation of the Sp(N) model. Peaks of exponentially increasing heights appear on the negative $b^0$-axis, rendering the wave function naively non-normalizable.
• Figure 4.1: Samples of, from top to bottom, $H_{xy}$, $B_{xy}=H_{xy}-\langle H_{xy}\rangle$, $\tilde{\beta}(x) = B_{xx}$ on 1d circle of size $L$ and lattice spacing $\frac{L}{K}$, $K=500$, with ${\cal D}$ = discrete Laplacian plus small mass term $\sim \frac{1}{L}$. From left to right, $N=1,10,250,10000$. $H$ for $N=1,10$ has reduced rank $2N$.
• Figure 5.1: Wave function squared $|\tilde{\psi}_0(b^0)|^2$ for $N=2$ (left) and $N=20$ (right). The positivity constraint on ${\cal H}={\cal D}+{\cal B}$ restricts $b^0>-\frac{3}{4}$, unlike in the naive version shown in fig. \ref{['fig:spnwave']}.
• Figure 5.2: Probability distribution $P(B_0)$ for $N=2$ (left) and $N=20$ (right). (For comparison to fig. \ref{['fig:psi0b']}, recall that $B_0 = \frac{\pi^2}{4} b^0 \approx 2.5 \, b^0$.)