Rodriguez (JHEP 2023) constructed a solvable string-theoretic cosmology whose worldsheet theory is Liouville conformal field theory at central charge c=1 for time, coupled to Liouville theory at c=25 for space, and showed that string perturbation theory computes cosmological wavefunction components, i.e. overlaps between an empty-looking 'Big Bang' initial state in the infinite past and out-states of massless closed strings in the infinite future. The computed components are strikingly simple: the tree-level three-point component equals 8 gs3 CS2ω1ω2ω38\,g_s^3\,C_{S^2}\omega_1\omega_2\omega_3, and the four-point component takes the form gs4 CS2ω1ω2ω3ω4(α+β∑iωi2)g_s^4\,C_{S^2}\omega_1\omega_2\omega_3\omega_4(\alpha+\beta\sum_i\omega_i^2) with the coefficients later sharpened numerically to exactly α=8\alpha=8, β=16\beta=16; the sphere normalization CS2C_{S^2} is undetermined. By analogy with the c=1 and type 0B strings, whose exact nonperturbative definition is a matrix quantum mechanics (MQM), this simplicity strongly suggests a holographic dual — most plausibly a time-dependent background of the c=1 MQM — but as of 2026 no dual has been identified, the initial state's nature (e.g. Hartle–Hawking-type or not) is unestablished, and unitarity/factorization constraints on cosmological overlaps and their normalization constants are unknown. The problem is to find this nonperturbative description and use it to derive, rather than merely recompute, the cosmological wavefunctions.

Background

In string perturbation theory, scattering amplitudes are computed as integrals of correlators of a two-dimensional conformal field theory (the 'worldsheet' CFT, one factor of which is Liouville CFT — an interacting CFT of a field with an exponential potential, whose central charge cc counts its degrees of freedom) over moduli spaces of Riemann surfaces, organized as a genus expansion in the string coupling gsg_s. Two-dimensional (2d) string theory, whose target spacetime has one time and one space dimension, is the oldest solvable laboratory for string-theoretic quantum gravity. Its best-studied instance, the c=1 string, has a matter sector consisting of a timelike free boson coupled to c=25 Liouville theory; its nonperturbative definition is a matrix quantum mechanics: the singlet sector of a large Hermitian matrix model in an inverted harmonic-oscillator potential, which after a 'double scaling' limit (tuning matrix size and coupling together) becomes NN free fermions filling one side of the inverted potential up to a Fermi level μ=1/(2πgs)\mu=1/(2\pi g_s). Fluctuations of the fermion density ('collective field') reproduce the spacetime massless closed-string 'tachyon' — the only propagating mode — and the matrix correlators reproduce the worldsheet string amplitudes, making this an exact holographic duality (Balthazar–Rodriguez–Yin 2019 and predecessors).

Within this framework Rodriguez (2023) constructed a solvable string cosmology. The worldsheet CFT is Liouville theory at c=1c=1 describing the time coordinate χ0\chi_0 (with a time-dependent Liouville potential) coupled to Liouville theory at c=25c=25 describing space, plus the reparametrization ghosts — one of very few exactly solvable worldsheet theories for a genuinely time-dependent string background. In the infinite past the time-dependent potential grows without bound, so worldsheets cannot propagate there and the spacetime looks 'empty' near a big-bang-type beginning (labeled 'Big Bang'); in the infinite future the potential vanishes and standard perturbative closed-string states of energies ωi>0\omega_i>0 exist. Since no ordinary S-matrix exists in such a background, string diagrams were interpreted as computing cosmological wavefunction components, Ψ(ω1,…,ωn)=⟨ω1,…,ωn∣Big Bang⟩\Psi(\omega_1,\dots,\omega_n)=\langle\omega_1,\dots,\omega_n|\text{Big Bang}\rangle: overlaps between the implicitly defined initial state and out-states in the far-future Hilbert space. Using the exact Liouville three-point coefficients and numerical integration of the Virasoro conformal-block decompositions of the Liouville four-point functions (conformal blocks are the standard building blocks into which CFT correlators decompose, each summing the descendants of one primary state) over the sphere moduli space, the paper found remarkably simple tree-level results: the three-point component Ψ(ω1,ω2,ω3)=8 gs3 CS2 ω1ω2ω3\Psi(\omega_1,\omega_2,\omega_3)=8\,g_s^3\,C_{S^2}\,\omega_1\omega_2\omega_3, with CS2C_{S^2} an undetermined normalization constant, and the four-point component gs4 CS2 ω1ω2ω3ω4(α+β∑iωi2)g_s^4\,C_{S^2}\,\omega_1\omega_2\omega_3\omega_4(\alpha+\beta\sum_i\omega_i^2) with fitted α=7.513±0.001\alpha=7.513\pm0.001, β=15.99±0.01\beta=15.99\pm0.01, sharpened to the exact values α=8\alpha=8, β=16\beta=16 by a subsequent more precise calculation of Collier–Eberhardt–Muehlmann–Rodriguez. A follow-up paper computed the torus (genus-one) one-point diagram numerically, again finding a simple form, and showed that the genus expansion organizes the wavefunction as a squeezed state of the massless particles, familiar from quantum field theory in time-dependent backgrounds.

This simplicity strongly suggests, by analogy with the c=1 and type 0B strings, that the cosmology admits a nonperturbative holographic dual — most plausibly a specific time-dependent background (a 'quantum quench' of the Fermi sea) of the c=1 matrix quantum mechanics. Time-dependent Fermi seas of the c=1 MQM have been studied from the matrix side since the early 2000s (Alexandrov–Kazakov–Kostov; Karczmarek–Strominger's 'matrix cosmology'), and recent work has analyzed the class of superluminal-Liouville-wall backgrounds, to which this cosmology belongs, directly in the matrix model. However, no matrix-model construction has ever been matched to this (or any comparable) solvable cosmological worldsheet theory. In time-independent backgrounds, perturbative unitarity — the factorization of amplitudes on lower-genus ones — fixes constants like CS2C_{S^2}; for cosmological overlaps the analogue is not understood. Nonperturbative D-instanton effects (contributions of order e−1/gse^{-1/g_s} from Euclidean D-branes, nonperturbative extended objects of string theory) are likewise uncomputed here. The initial state also resembles, but has not been shown to be, a Hartle–Hawking-type 'no-boundary' state — the state selected by replacing the Lorentzian past of a cosmology by a Euclidean cap so that the universe emerges from 'nothing'.

Problem Statement

Find the nonperturbative (holographic) description of the two-dimensional string cosmology of Rodriguez (2023) — the solvable worldsheet theory of Liouville CFT at c=1c=1 for time coupled to Liouville CFT at c=25c=25 for space — whose single overarching objective is to explain, rather than merely recompute, the perturbatively calculated cosmological wavefunctions Ψ(ω1,…,ωn)=⟨ω1,…,ωn∣Big Bang⟩\Psi(\omega_1,\dots,\omega_n)=\langle\omega_1,\dots,\omega_n|\text{Big Bang}\rangle. A complete answer should achieve the following jointly linked results: (1) identify the matrix quantum mechanics — expected to be a specific time-dependent background or solution of the c=1 matrix quantum mechanics — whose correlators reproduce the computed wavefunction components, in particular deriving analytically the tree-level three-point component 8 gs3 CS2 ω1ω2ω38\,g_s^3\,C_{S^2}\,\omega_1\omega_2\omega_3 and the four-point component gs4 CS2 ω1ω2ω3ω4(8+16∑iωi2)g_s^4\,C_{S^2}\,\omega_1\omega_2\omega_3\omega_4(8+16\sum_i\omega_i^2), and explaining the origin of the simple quadratic-in-energies structure and integer coefficients; (2) characterize the 'Big Bang' initial state implicit in the worldsheet construction — in particular whether it is a Hartle–Hawking-type empty/no-boundary state — its realization in the dual, and how it deforms under generic marginal deformations of the worldsheet CFT (continuous operator deformations of the theory that preserve conformal invariance); and (3) determine what unitarity and factorization imply for cosmological overlaps, fixing (or proving unfixed) the sphere normalization constant CS2C_{S^2} and its genus-one (torus) analogue CT2C_{T^2}.

References

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