Quantum geometry of topological gravity (original) (raw)

Dynamical triangulations, a gateway to quantum gravity?

Journal of Mathematical Physics, 1995

We show how it is possible to formulate Euclidean two-dimensional quantum gravity as the scaling limit of an ordinary statistical system by means of dynamical triangulations, which can be viewed as a discretization in the space of equivalence classes of metrics. Scaling relations exist and the critical exponents have simple geometric interpretations. Hartle-Hawkings wave functionals as well as reparametrization invariant correlation functions which depend on the geodesic distance can be calculated. The discretized approach makes sense even in higher dimensional space-time. Although analytic solutions are still missing in the higher dimensional case, numerical studies reveal an interesting structure and allow the identi cation of a xed point where we can hope to de ne a genuine non-perturbative theory of four-dimensional quantum gravity.

Intrinsic geometric structure of c = −2 quantum gravity

Nuclear Physics B - Proceedings Supplements, 1998

We couple c = ?2 matter to 2-dimensional gravity within the framework of dynamical triangulations. We use a very fast algorithm, special to the c = ?2 case, in order to test scaling of correlation functions de ned in terms of geodesic distance and we determine the fractal dimension dH with high accuracy. We nd dH = 3:58(4), consistent with a prediction coming from the study of di usion in the context of Liouville theory, and that the quantum space{time possesses the same fractal properties at all distance scales similarly to the case of pure gravity.

On the fractal structure of two-dimensional quantum gravity

1995

We provide evidence that the Hausdor dimension is 4 and the spectral dimension is 2 for two-dimensional quantum gravity coupled the matter with a central charge c 1. For c > 1 the Hausdor dimension and the spectral dimension monotonously decreases to 2 and 1, respectively.

The quantum spacetime of c > 0 2d gravity

Nuclear Physics B - Proceedings Supplements, 1998

We review recent developments in the understanding of the fractal properties of quantum spacetime of 2d gravity coupled to c > 0 conformal matter. In particular we discuss bounds put by numerical simulations using dynamical triangulations on the value of the Hausdorff dimension dH obtained from scaling properties of two point functions defined in terms of geodesic distance. Further insight to the fractal structure of spacetime is obtained from the study of the loop length distribution function which reveals that the 0 < c ≤ 1 system has similar geometric properties with pure gravity, whereas the branched polymer structure becomes clear for c ≥ 5.

Geometry of a two-dimensional quantum gravity: Numerical study

Nuclear Physics B, 1991

A two-dimensional quantum gravity is simulated by means of the dynamical triangulation model. The size of the lattice was up to hundred thousand triangles. Massively parallel simulations and recursive sampling were implemented independently and produced similar results. Wherever the analytical predictions existed, our results confirmed them. The cascade process of baby universes formulation a la Coleman-Hawking scenario in a two-dimensional case has been observed. We observed that there is a simple universal inclusive probability for a baby universe to appear. This anomalous branching of surfaces led to a rapid growth of the integral curvature inside a circle. The volume of a disk in the internal metric has been proven to grow faster than any power of radius. The scaling prediction for the mean square extent given by the Liouville theory has been confirmed. However, the naive expectation for the average Liouville lagrangian < f (p~6)2 > is about 1 order of magnitude different from the results. This apparently points out to some flaws in the current definition of a Liouville model.

The spectral dimension of 2D quantum gravity

Journal of High Energy Physics, 1998

We show that the spectral dimension d s of two-dimensional quantum gravity coupled to Gaussian fields is two for all values of the central charge c ≤ 1. The same arguments provide a simple proof of the known result d s = 4/3 for branched polymers.

The toroidal Hausdorff dimension of 2d Euclidean quantum gravity

Physics Letters B, 2013

The lengths of shortest non-contractible loops are studied numerically in 2d Euclidean quantum gravity on a torus coupled to conformal field theories with central charge less than one. We find that the distribution of these geodesic lengths displays a scaling in agreement with a Hausdorff dimension given by the formula of Y. Watabiki.

On Finite Size Effects in d=2d=2d=2 Quantum Gravity

A systematic investigation is given of finite size effects in d = 2 quantum gravity or equivalently the theory of dynamically triangulated random surfaces.For Ising models coupled to random surfaces, finite size effects are studied on the one hand by numerical generation of the partition function to arbitrary accuracy by a deterministic calculus, and on the other hand by an analytic theory based on the singularity analysis of the explicit parametric form of the free energy of the corresponding matrix model. Both these reveal that the form of the finite size corrections, not surprisingly, depend on the string susceptibility.For the general case where the parametric form of the matrix model free energy is not explicitly known, it is shown how to perform the singularity analysis.All these considerations also apply to other observables like susceptibility etc.In the case of the Ising model it is shown that the standard Fisher-scaling laws are reproduced. A study of finite size effects in statistical systems is of importance for a variety of reasons. It is indispensable for a more reliable interpretation of numerical data. As shown by Cardy in the context of systems with conformal invariance, finite size effects also codify the spectrum of the theory. In most cases finite size effects are estimated or parametrised by extrapolating the results of simulations carried out for various sizes. It is the purpose of this article to show how such finite size effects for two dimensdional quantum gravity systems can be studied systematically. Let us begin with the case of so called "pure gravity" which is mapped

Common structures in simplicial quantum gravity

Physics Letters B - PHYS LETT B, 1999

Statistical properties of dynamically triangulated manifolds (DT mfds) in terms of the geodesic distance are studied numerically. The string susceptibility exponents for the boundary surfaces in three-dimensional DT mfds are measured numerically. For spherical boundary surfaces, we obtain a result consistent with the case of a two-dimensional spherical DT surface described by the matrix model. This gives a correct method to reconstruct two-dimensional random surfaces from three-dimensional DT mfds. Furthermore, a scaling property of the volume distribution of minimum neck baby universes is investigated numerically in the case of three and four dimensions, and we obtain a common scaling structure near to the critical points belonging to the strong coupling phase in both dimensions. We have evidence for the existence of a common fractal structure in three- and four-dimensional simplicial quantum gravity.

Quantum gravity, dynamical triangulations and higher-derivative regularization

Nuclear Physics B, 1993

We consider a discrete model of euclidean quantum gravity in four dimensions based on a summation over random simplicial manifolds. The action used is the Einstein-Hilbert action plus an R 2-term. The phase diagram as a function of the bare coupling constants is studied in the search for a sensible continuum limit. For small values of the coupling constant of the R 2 term the model seems to belong to the same universality class as the model with pure Einstein-Hilbert action and exhibits the same phase transition. The order of the transition may be second or higher. The average curvature is positive at the phase transition, which makes it difficult to understand the possible scaling relations of the model.