Contributor info: Dr Beatrix Mühlmann

Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann

SciPost Phys. 19, 115 (2025) · published 28 October 2025 | Toggle abstract · pdf

We give a rigorous definition of sine dilaton gravity in terms of the worldsheet theory of the complex Liouville string [S. Collier et al., Phys. Rev. Lett. 134, 251602 (2025)]. The latter has a known exact solution that we leverage to explore the gravitational path integral of sine dilaton gravity – a quantum deformation of dS JT gravity that admits both AdS$_2$ and dS$_2$ vacua. We uncover that the gravitational path integral receives contributions from new saddles describing transitions between vacua in a third-quantized picture. We also discuss the sphere and disk partition function in this context and contrast our findings with other recent work on this theory.

Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann, Victor A. Rodriguez

SciPost Phys. 19, 034 (2025) · published 12 August 2025 | Toggle abstract · pdf

We investigate general observables of the complex Liouville string with worldsheet boundaries. We develop a universal formalism that reduces such observables to ordinary closed string amplitudes without boundaries, applicable to any worldsheet string theory, but particularly simple in the context of 2d or minimal string theories. We apply this formalism to the duality of the complex Liouville string with the matrix integral proposed in [SciPost Phys. 19, 033 (2025); SciPost Phys. 18, 154 (2025)] and showcase the formalism by finding appropriate boundary conditions for various matrix model quantities of interest, such as the resolvent or the partition function. We also apply this formalism towards the computation of non-perturbative effects on the worldsheet mediated by ZZ-instantons. These are known to be plagued by extra subtleties which need input from string field theory to resolve. These computations probe and uncover the duality between the complex Liouville string and the matrix model at the non-perturbative level.

Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann, Victor A. Rodriguez

SciPost Phys. 19, 033 (2025) · published 12 August 2025 | Toggle abstract · pdf

We introduce a new two-dimensional string theory defined by coupling two copies of Liouville CFT with complex central charge $c=13± i \lambda$ on the worldsheet. This string theory defines a novel, consistent and controllable model of two-dimensional quantum gravity. We use the exact solution of the worldsheet theory to derive stringent constraints on the analytic structure of the string amplitudes as a function of the vertex operator momenta. Together with other worldsheet constraints, this allows us to completely pin down the string amplitudes without explicitly computing the moduli space integrals. We focus on the case of the sphere four-point amplitude and torus one-point amplitude as worked examples. This is the first in a series of papers on the complex Liouville string: three subsequent papers will elucidate the holographic duality with a two-matrix integral, discuss worldsheet boundaries and non-perturbative effects, and connect the theory to de Sitter quantum gravity.

Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann, Victor A. Rodriguez

SciPost Phys. 18, 154 (2025) · published 13 May 2025 | Toggle abstract · pdf

We propose a duality between the complex Liouville string and a two-matrix integral. The complex Liouville string is defined by coupling two Liouville theories with complex central charges $c = 13 ± i \lambda$ on the worldsheet. The matrix integral is characterized by its spectral curve which allows us to compute the perturbative string amplitudes recursively via topological recursion. This duality constitutes a controllable instance of holographic duality. The leverage on the theory is provided by the rich analytic structure of the string amplitudes that we discussed in [arXiv:2409.18759] and allows us to perform numerous tests on the duality.

Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann

SciPost Phys. 18, 131 (2025) · published 17 April 2025 | Toggle abstract · pdf

We propose a precise duality between pure de Sitter quantum gravity in $2+1$ dimensions and a double-scaled matrix integral. This duality unfolds in two distinct aspects. First, by carefully quantizing the gravitational phase space, we arrive at a novel proposal for the quantum state of the universe at future infinity. We compute cosmological correlators of massive particles in the universe specified by this wavefunction. Integrating these correlators over the metric at future infinity yields gauge-invariant observables, which are identified with the string amplitudes of the complex Liouville string [S. Collier et al., arXiv: 2409.17246]. This establishes a direct connection between integrated cosmological correlators and the resolvents of the matrix integral dual to the complex Liouville string, thereby demonstrating one aspect of the dS$_3$/matrix integral duality. The second aspect concerns the cosmological horizon of the dS static patch and the Gibbons-Hawking entropy it is conjectured to encode. We show that this entropy can be reproduced exactly by counting the entries of the matrix.

Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann, Victor A. Rodriguez

SciPost Phys. 16, 057 (2024) · published 26 February 2024 | Toggle abstract · pdf

We introduce a critical string theory in two dimensions and demonstrate that this theory, viewed as two-dimensional quantum gravity on the worldsheet, is equivalent to a double-scaled matrix integral. The worldsheet theory consists of Liouville CFT with central charge c$≥$ 25 coupled to timelike Liouville CFT with central charge 26-c. The double-scaled matrix integral has as its leading density of states the universal Cardy density of primaries in a two-dimensional CFT, thus motivating the name Virasoro minimal string. The duality holds for any value of the continuous parameter $c$ and reduces to the JT gravity/matrix integral duality in the large central charge limit. It thus provides a precise stringy realization of JT gravity. The main observables of the Virasoro minimal string are quantum analogues of the Weil-Petersson volumes, which are computed as absolutely convergent integrals of worldsheet CFT correlators over the moduli space of Riemann surfaces. By exploiting a relation of the Virasoro minimal string to three-dimensional gravity and intersection theory on the moduli space of Riemann surfaces, we are able to give a direct derivation of the duality. We provide many checks, such as explicit numerical - and in special cases, analytic - integration of string diagrams, the identification of the CFT boundary conditions with asymptotic boundaries of the two-dimensional spacetime, and the matching between the leading non-perturbative corrections of the worldsheet theory and the matrix integral. As a byproduct, we discover natural conformal boundary conditions for timelike Liouville CFT.

Ankit Aggarwal, Alejandra Castro, Stéphane Detournay, Beatrix Mühlmann

SciPost Phys. 15, 083 (2023) · published 7 September 2023 | Toggle abstract · pdf

A holographic description of three-dimensional warped black holes suffers from ambiguities due to a seemingly harmless choice of coordinate system. This gives rise to the notion of ensembles in warped black holes, and we focus on two of them: the canonical and quadratic ensemble. Our aim is to quantify the imprint of these ensembles in the near-extremal limit of a warped black hole. To this end, for each ensemble, we explore the thermodynamic response and evaluate greybody factors. We also set-up a holographic dictionary in their near-AdS$_2$ region, and decode aspects of the dual near-CFT$_1$. This gives us different perspectives of the black hole that we can contrast and compare. On the one hand, we find perfect agreement between the near-extremal limit of the canonical ensemble warped black holes, their near-AdS$_2$ effective analysis, and a warped conformal field theory description. On the other, we are led to rule out the quadratic ensemble due to inconsistencies at the quantum level with the near-AdS$_2$ effective description.

Ankit Aggarwal, Alejandra Castro, Stéphane Detournay, Beatrix Mühlmann

SciPost Phys. 15, 056 (2023) · published 9 August 2023 | Toggle abstract · pdf

Warped conformal field theories (WCFTs) are two-dimensional non-relativistic systems, with a chiral scaling and shift symmetry. We present a detailed derivation of the near-extremal limit for their torus partition function. This limit requires large values of the central charge, and is only consistent for non-unitary WCFT. We compare our analysis with previous studies of WCFT and its relation to a one-dimensional warped-Schwarzian theory. We discuss different ensembles of warped CFTs and contrast our results with analogous limits in two-dimensional CFTs.