Quantization and non-perturbative duals of non-relativistic and Carrollian Jackiw–Teitelboim gravity

Jackiw–Teitelboim (JT) gravity in two dimensions admits well-defined non-relativistic ($c\to\infty$) and Carrollian ($c\to0$) limits, whose classical bulk actions and Schwarzian-type boundary actions were constructed by Gomis, Hidalgo and Salgado-Rebolledo. The open problem is the quantum completion of this program: quantize the non-relativistic (NR) and Carrollian Schwarzian boundary theories — partition function, density of states, correlation functions, and path-integral measure — determine whether these theories admit non-perturbative completions as double-scaled matrix integrals along the lines of the JT-gravity-as-matrix-integral correspondence, and identify a concrete regime of a SYK-like quantum-mechanical model whose low-energy effective dynamics is the NR or Carrollian Schwarzian. The full quantization, matrix-model duals, and SYK identification remain open.

Background

Jackiw–Teitelboim (JT) gravity is a two-dimensional dilaton gravity theory with negative cosmological constant that plays a central role in solvable holography: its boundary dynamics is governed by the Schwarzian action, it captures the low-energy physics of the Sachdev–Ye–Kitaev (SYK) model — a solvable disordered model of quantum mechanics — and Saad, Shenker and Stanford showed that JT gravity is non-perturbatively dual to a double-scaled random matrix integral, i.e. an ensemble of matrices whose genus-by-genus expansion reproduces the topological expansion of the gravitational path integral. In the first-order formulation JT gravity is a BF theory (a topological gauge theory whose action couples a gauge field $A$ to an auxiliary field $B$ in the adjoint representation); to obtain finite limits as the speed of light $c$ tends to $\infty$ (non-relativistic, NR) or to $0$ (Carrollian), the gauge algebra must be enlarged from $\mathfrak{sl}(2,\mathbb{R})$ to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$. The resulting NR bulk theory has Newton–Hooke gauge symmetry (the Galilei algebra with a cosmological constant) and lives on Newton–Cartan geometry — a non-Lorentzian geometry in which the Lorentzian metric is replaced by a degenerate spatial metric together with a clock one-form — while the Carrollian theory has (A)dS-Carroll gauge symmetry and lives on Carroll geometry, the ultra-relativistic dual in which a degenerate temporal metric replaces the spatial one and there is no lightcone propagation. The boundary actions are obtained by the method of non-linear realisations: spacetime symmetries are spontaneously broken on a coset, redundant Goldstone fields are eliminated by inverse Higgs constraints, and the remaining boundary modes are governed by a particle-on-a-group action. For JT this reproduces the Drinfeld–Sokolov reduction — a Hamiltonian reduction procedure on the group SL(2,R) — yielding the Schwarzian action. Gomis, Hidalgo and Salgado-Rebolledo (2021) applied this construction to the NR and Carrollian cases, obtaining the NR Schwarzian action (their eq. (6.8)) and its Carrollian counterpart, deriving the conserved charges and finding the asymptotic symmetry algebra to be the twisted warped Virasoro algebra — a Virasoro algebra combined with a $\mathfrak{u}(1)$ current algebra with a twisting term — and reconstructing the Newton–Cartan and Carrollian bulk geometry from the boundary data. What the classical construction does not provide is the quantum theory: the exact partition function and density of states of the NR/Carrollian Schwarzian theories (including the correct path-integral measure, which is not fixed by the standard coadjoint-orbit construction because the symmetry algebra is non-semisimple), any matrix-model or other non-perturbative ensemble description, and any microscopic matter realization. The motivation stated in the source paper is to find another physical sector of the SYK model and its holographic description: just as the ordinary Schwarzian is the common low-energy theory of JT gravity and SYK, the NR and Carrollian Schwarzians are candidate low-energy theories of novel NR/ultra-relativistic sectors of SYK-like models, a role made plausible by the known relation between complex SYK, warped conformal field theories, and warped-Schwarzian-type boundary actions.

Problem Statement

Establish the quantum theory of the non-relativistic and Carrollian limits of Jackiw–Teitelboim gravity and their Schwarzian-type boundary actions, and identify their non-perturbative holographic duals. Concretely: (1) Quantize the NR Schwarzian boundary theory (eq. (6.8) of Gomis–Hidalgo–Salgado-Rebolledo) and its Carrollian counterpart (the boundary actions of eqs. (6.27) and (6.45) in the same paper, which are shown there to admit the interpretation of boundary actions of Carrollian JT gravity) — compute the partition function, density of states, and correlation functions, together with a derivation of the correct path-integral measure; the standard coadjoint-orbit construction (quantization on the orbit of the symmetry group equipped with its natural Kirillov–Kostant symplectic form, which fixes the measure in the ordinary Schwarzian case) does not determine the measure for the non-semisimple twisted warped Virasoro symmetry algebra, and the result must go beyond the one-loop results available for the non-relativistic de Sitter case. (2) Determine whether, by analogy with the realization of JT gravity as a double-scaled matrix integral (Saad–Shenker–Stanford), the NR and Carrollian JT theories admit matrix-model (or other exactly solvable ensemble) non-perturbative completions whose genus expansions reproduce their boundary amplitudes — or prove that no such completion exists. (3) Identify a concrete regime of a SYK-like quantum-mechanical model whose effective low-energy dynamics is the NR (respectively Carrollian) Schwarzian, matching the symmetry-breaking pattern, spectrum, correlation functions, and path-integral measure. These three deliverables serve one common objective — the non-perturbative quantum theory of NR/Carrollian JT gravity and its holographic interpretation.

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