Model Of Two-dimensional Quantum Gravity Research Articles

Critical JT gravity

In this paper, we investigate a critical behavior of JT gravity, a model of two-dimensional quantum gravity on constant negatively curved spacetimes. Our approach involves using techniques from random maps to investigate the generating function of Weil-Petersson volumes, which count random hyperbolic surfaces with defects. The defects are weighted geodesic boundaries, and criticality is reached by tuning the weights to the regime where macroscopic holes emerge in the hyperbolic surface, namely non-generic criticality. We analyze the impact of this critical regime on some universal features, such as its density of states. We present a family of models that interpolates between systems with ρ0(E) ~ sqrt{E-{E}_0} and ρ0(E) ~ (E − E0)3/2, which are commonly found in models of JT gravity coupled to dynamical end-of-the-world and FZZT branes, and give a precise definition of what this phase transition means from the random geometry point of view.

Replica wormholes from Liouville theory

The replica wormholes are a key to the existence of the islands that play a central role in a recent proposal for the resolution of the black hole information paradox. In this paper, we study the replica wormholes in the JT gravity, a model of two-dimensional quantum gravity coupled to a non-dynamical dilaton, by making use of the 2d conformal field theory (CFT) description, namely, the Liouville theory coupled to the (2, p) minimal matter in the p → ∞ limit. In the Liouville CFT description, the replica wormholes are created by the twist operators and the gravitational part of the bulk entanglement entropy can be reproduced from the twist operator correlators. We propose the precise dictionary and show how this correspondence works in detail.

Duality of 2D Gravity as a Local Fourier Duality

The p - q duality is a relation between the (p,q) model and the (q,p) model of two-dimensional quantum gravity. Geometrically this duality corresponds to a relation between the two relevant points of the Sato Grassmannian. Kharchev and Marshakov have expressed such a relation in terms of matrix integrals. Some explicit formulas for small p and q have been given in the work of Fukuma-Kawai-Nakayama. Already in the duality between the (2,3) model and the (3,2) model the formulas are long. In this work a new approach to p - q duality is given: It can be realized in a precise sense as a local Fourier duality of D-modules. This result is obtained as a special case of a local Fourier duality between irregular connections associated to Kac-Schwarz operators. Therefore, since these operators correspond to Virasoro constraints, this allows to view the p - q duality as a consequence of the duality of the relevant Virasoro constraints.

Trees and spatial topology change in causal dynamical triangulations

Generalized causal dynamical triangulations (generalized CDT) is a model of two-dimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur. We solve the model at the discretized level using bijections between quadrangulations and trees. In the continuum limit (scaling limit) the amplitudes are shown to agree with known formulas and explicit expressions are obtained for loop propagators and two-point functions. It is shown that from a combinatorial point of view generalized CDT can be viewed as the scaling limit of planar maps with a finite number of faces and we determine the distance function on this ensemble of planar maps. Finally, the relation with planar maps is used to illuminate the mysterious identity of certain continuum cylinder amplitudes.

On the Spectral Dimension of Causal Triangulations

We introduce an ensemble of infinite causal triangulations, called the uniform infinite causal triangulation, and show that it is equivalent to an ensemble of infinite trees, the uniform infinite planar tree. It is proved that in both cases the Hausdorff dimension almost surely equals 2. The infinite causal triangulations are shown to be almost surely recurrent or, equivalently, their spectral dimension is almost surely less than or equal to 2. We also establish that for certain reduced versions of the infinite causal triangulations the spectral dimension equals 2 both for the ensemble average and almost surely. The triangulation ensemble we consider is equivalent to the causal dynamical triangulation model of two-dimensional quantum gravity and therefore our results apply to that model.

On the Asymptotic Number of Plane Curves and Alternating Knots

We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of plane curves, a similar conjecture is made for the asymptotic number of prime alternating knots. The rationale leading to these conjectures is given by quantum field theory. Plane curves are viewed as configurations of loops on random planar lattices, that are in turn interpreted as a model of two-dimensional quantum gravity with matter. The identification of the universality class of this model yields the conjecture. Since approximate counting or sampling planar curves with more than a few dozens of intersections is an open problem, direct confrontation with numerical data yields no convincing indication on the correctness of our conjectures. However, our physical approach yields a more general conjecture about connected systems of curves. We take advantage of this to design an original and feasible numerical test, based on recent perfect samplers for large planar maps. The numerical data strongly support our identification with a conformal field theory recently described by Readand Saleur.

Model of two-dimensional quantum gravity in the strong coupling regime.

A model of two-dimensional quantum gravity that is the analog of the tensionless string is proposed. The gravitational constant ($k$) is the analog of the Regge slope ($\alpha^{'}$) and it shows that when $k \rightarrow \infty$, $2D$ quantum gravity can be understood as a tensionless string theory embeded in a two-dimensional target space. The temporal coordinate of the target space play the role of time and the wave function can be interpreted as in standard quantum mechanics.

A modified c = 1 matrix model with new critical behavior

By introducing a ∫ dt g (Tr Φ 2( t)) 2 term into the action of the c = 1 matrix model of two-dimensional quantum gravity, we find a new critical behavior for random surfaces. The planar limit of the path integral generates multiple spherical “bubbles” which touch one another at single points. At a special value of g, the sum over connected surfaces behaves as Δ 2log Δ, where Δ is the cosmological constant (the sum over surfaces of area A goes as A −3). For comparison, in the conventional c = 1 model the sum over planar surfaces behaves as Δ 2 log Δ .

COMBINATORIAL AND TOPOLOGICAL PHASE STRUCTURE OF NON-PERTURBATIVE n-DIMENSIONAL QUANTUM GRAVITY

We provide a non-perturbative geometrical characterization of the partition function of ndimensional quantum gravity based on a rough classification of Riemannian geometries. We show that, under natural geometrical constraints, the theory admits a continuum limit with a non-trivial phase structure parametrized by the homotopy types of the class of manifolds considered. The results obtained qualitatively coincide, when specialized to dimension two, with those of two-dimensional quantum gravity models based on random triangulations of surfaces.

Symmetries of integrable hierarchies and matrix model constraints

The orbit construction associates a soliton hierarchy to every level-one vertex realization of a simply laced affine Kac-Moody algebra g. We show that the τ-function of such a hierarchy has the (truncated) Virasoro algebra as an algebra of infinitesimal symmetry transformations. To prove this we use an appropriate bilinear form of these hierarchies together with the coset construction of conformal field theory. For A l (1) the orbit construction gives either the Toda or the KdV hierarchy. These both occur in the one-matrix model of two-dimensional quantum gravity, before and after the double scaling limit respectively. The truncated Virasoro symmetry algebra is exactly the algebra of constraints of the one-matrix model. The partition function of the one-matrix model is therefore an invariant τ-function. We also consider the case of A l (1) with l > 1. Surprisingly, the symmetry algebra in that case is not simply a truncated Casimir algebra. It appears that again only the Virasoro symmetry survives. We speculate on the relation with multi-matrix models.

Bounded geometries and topology fluctuations in lattice quantum gravity

The authors exploit the coarse classification of Riemannian geometries provided by Gromov's pre-compactness theorem (1981) in order to give a rigorous characterization of the partition function of n-dimensional, (n>or=2), lattice quantum gravity. They prove that the resulting theory admits a continuum limit describing phase transitions between different homotopy types of manifolds, with phases parametrized by the fundamental group and by the (Whitehead) torsions of the manifolds sampled. They also show that the results obtained coincide, when specialized to dimension two, with those of two-dimensional quantum gravity models based on random triangulations of surfaces.

Ground rings and their modules in 2D gravity with c⩽1 matter

All solvable two-dimensional quantum gravity models have non-trivial BRST cohomology with vanishing ghost number. These states from a ring and all the other states in the theory fall into modules of this ring. The relations in the ring and in the modules have a physical interpretation. The existence of these rings and modules leads to non-trivial constraints on the correlation functions and goes a long way toward solving these theories in the continuum approach.

Do integrable mappings have the Painlevé property?

We present an integrability criterion for discrete-time systems that is the equivalent of the Painlev\'e property for systems of a continuous variable. It is based on the observation that for integrable mappings the singularities that may appear are confined, i.e., they do not propagate indefinitely when one iterates the mapping. Using this novel criterion we show that there exists a family of nonautonomous integrable mappings which includes the discrete Painlev\'e equations by ${\mathit{P}}_{\mathrm{I}}$, recently derived in a model of two-dimensional quantum gravity, and ${\mathit{P}}_{\mathrm{II}}$, obtained as a similarity reduction of the integrable modified Korteweg--de Vries lattice. These systems possess Lax pairs, a well-known integrability feature.

Simplicial quantum gravity in more than two dimensions.

The dynamical triangulation (DT) model of two-dimensional quantum gravity is generalized to higher dimensions. In three dimensions two updates are described which appear to allow for ergodic sampling of states, and an updating procedure which obeys detailed balance is given. The tensor-model generalization of the matrix-model formulation of higher-dimensional gravity is discussed and the relations between DT-, continuum-, and tensor-model couplings are given.