Matrix and Lower-Dimensional de Sitter Models

Lower-dimensional de Sitter models can define exact boundary amplitudes, moduli integrals, and genus expansions that are inaccessible in four-dimensional gravity. Their value comes from precisely stated degrees of freedom and contours. A matrix-like completion may describe an ensemble, a resummation, or a formal continuation with no positive measure; none of those possibilities may be promoted to universal four-dimensional cosmological holography.

Required background. de Sitter and Cosmological Holography supplies the proposal taxonomy; Nonperturbative Definition and Completion Criteria supplies the fixed-theory test.

Helpful background. Fixed-Theory Factorization and Nonperturbative Completion Tests treats ensemble ambiguity; IKKT Type-IIB Matrix Model and Emergent-Spacetime Claims gives the distinct matrix-spacetime standard.

de Sitter JT as a defined laboratory

The Lorentzian dS$_2$ JT action can be written schematically as

$$I_{\mathrm{dS\,JT}} =\frac{S_0}{4\pi}\int\sqrt{-g}\,R +\frac12\int\sqrt{-g}\,\Phi(R-2H^2)+I_{\partial},$$

with boundary terms and an integration contour chosen for the desired expanding or contracting asymptotics. The dilaton equation fixes locally de Sitter curvature, while boundary reparametrization modes carry the nontrivial dynamics. One can define Hartle–Hawking wavefunctions, future-to-past amplitudes, or scattering-like objects; these are not the same observable.

At fixed topology, the path integral reduces to moduli integrals. For an orientable genus-$g$ surface with $n$ asymptotic boundaries, $\chi=2-2g-n$, and the Euler term supplies the factor $e^{S_0\chi}$. Thus a disk carries $e^{S_0}$, while each additional handle is suppressed by $e^{-2S_0}$; the opposite sign would reverse the semiclassical topology hierarchy. Cotler, Jensen, and Maloney related early dS JT amplitudes to analytic continuations of AdS JT and matrix-model structures Cotler, Jensen, and Maloney 2020, §§ 3–6. The continuation changes convergence and reality properties.

First application: topology expansion and late-time data

Let $\mathcal A(\ell_+,\ell_-)$ denote an amplitude between specified asymptotic boundary lengths. Its connected topology expansion has the form

$$\mathcal A(\ell_+,\ell_-) \sim\sum_{g\ge0}g_{\mathrm{eff}}^{2g-2+n}\, \mathcal V_{g,n}(\ell_+,\ell_-),$$

where $n$ counts boundaries and $\mathcal V_{g,n}$ is the continued moduli-space integral. The magnitude $\lvert g_{\mathrm{eff}}\rvert=e^{-S_0}$ reproduces $e^{S_0\chi}$; in the nonperturbative dS JT continuation, $g_{\mathrm{eff}}=i e^{-S_0}$ is imaginary. The resulting alternating genus series is compatible with Borel–Le Roy resummation and can be represented formally by a matrix integral with a negative number of degrees of freedom Cotler and Jensen 2024, §§ 2–5.

This is a concrete result: specified amplitudes have a calculable asymptotic expansion with unusual but constrained resurgent behavior. It is not automatically a conventional Hermitian ensemble with positive spectral density. A resummation prescription or integration cycle is part of the nonperturbative definition, and different cycles can share the same perturbative saddle expansion.

The late-time limit must also be typed. A boundary length, dilaton value, or asymptotic momentum in dS JT is not a four-dimensional curvature perturbation. Factorization of a fixed boundary quantum mechanics, ensemble averaging in a matrix integral, and wormhole contributions are competing interpretations until a microscopic model selects one.

Matrix models are not one category

A double-scaled random matrix integral exactly defines averaged spectral observables once its potential and contour are fixed. A Lorentzian matrix model such as IKKT instead starts from matrices with a gauge symmetry and proposes emergent spacetime through particular states or saddles. A matrix representation of a topological expansion does not imply the latter microscopic dynamics. Degrees of freedom, observables, and large-$N$ limits must be compared explicitly.

Dimensional extrapolation is especially restricted. Two-dimensional dilaton gravity has no local graviton, while four-dimensional de Sitter has gravitons, gauge constraints, infrared effects, Kaluza–Klein and string sectors in any embedding, and metastability questions. Solving the former does not control the latter.

Adversarial control: preserve the saddle, change the completion

Choose two integration contours or matrix potentials with the same formal genus expansion around the leading de Sitter saddle but different exponentially small sectors. Compare a late-time amplitude or spectral observable. If they differ by $e^{-1/g_{\mathrm{eff}}}$ effects, the semiclassical series did not specify the completion. Then change topology weights while retaining the disk saddle; any claimed universal nonperturbative conclusion must be downgraded.

The evidence ceiling is exact and resurgent information in named lower-dimensional models, including explicit contour and ensemble questions. It is not a fixed four-dimensional theory, a generic dS/CFT dictionary, or evidence that all cosmological path integrals are matrix ensembles. Each result hands off with its Hamiltonian or contour, observable, genus parameter, and completion class intact.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Cotler, J., and Jensen, K. (2024). “Non-Perturbative de Sitter Jackiw–Teitelboim Gravity.” Journal of High Energy Physics 2024(12), 016. DOI.
• Cotler, J., Jensen, K., and Maloney, A. (2020). “Low-Dimensional de Sitter Quantum Gravity.” Journal of High Energy Physics 2020(6), 048. DOI.