Tensor network formulation of two-dimensional gravity

Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space

A set of physical operators which are responsible for touching interactions in the framework of c<1 unitary conformal matter coupled to 2D quantum gravity is found. As a special case the non-critical bosonic strings are considered. Some analogies with four dimensional quantum gravity are also discussed, e.g. creation-annihilation operators for baby universes, Coleman mechanism for the cosmological constant.

One can try to define the theory of quantum gravity as the sum over geometries. In two dimensions the sum over {\it Euclidean} geometries can be performed constructively by the method of {\it dynamical triangulations}. One can define a {\it proper-time} propagator. This propagator can be used to calculate generalized Hartle-Hawking amplitudes and it can be used to understand the the fractal structure of {\it quantum geometry}. In higher dimensions the philosophy of defining the quantum theory, starting from a sum over Euclidean geometries, regularized by a reparametrization invariant cut off which is taken to zero, seems not to lead to an interesting continuum theory. The reason for this is the dominance of singular Euclidean geometries. Lorentzian geometries with a global causal structure are less singular. Using the framework of dynamical triangulations it is possible to give a constructive definition of the sum over such geometries, In two dimensions the theory can be solved analytically. It differs from two-dimensional Euclidean quantum gravity, and the relation between the two theories can be understood. In three dimensions the theory avoids the pathologies of three-dimensional Euclidean quantum gravity. General properties of the four-dimensional discretized theory have been established, but a detailed study of the continuum limit in the spirit of the renormalization group and {\it asymptotic safety} is till awaiting.

We re-examine the nonperturbative curvature properties of two-dimensional Euclidean quantum gravity, obtained as the scaling limit of a path integral over dynamical triangulations of a two-sphere, which lies in the same universality class as Liouville quantum gravity. The diffeomorphism-invariant observable that allows us to compare the averaged curvature of highly quantum-fluctuating geometries with that of classical spaces is the so-called curvature profile. A Monte Carlo analysis on three geometric ensembles, which are physically equivalent but differ by the inclusion of local degeneracies, leads to new insights on the influence of finite-size effects. After eliminating them, we find strong evidence that the curvature profile of 2D Euclidean quantum gravity is best matched by that of a classical round four-sphere, rather than the five-sphere found in previous work. Our analysis suggests the existence of a well-defined quantum Ricci curvature in the scaling limit.

Operator product expansion in two-dimensional quantum gravity

It is shown that one can formulate Euclidean two-dimensional quantum gravity as the scaling limit of an ordinary statistical system. Scaling relations can be derived and the critical exponents have simple geometric interpretations. In addition it is possible to calculate reparametrization-invariant correlation functions as functions of the geodesic distance. The definitions for pure gravity generalize to matter systems coupled to two-dimensional quantum gravity. As long as the central charge c of the matter field is less than 1, it is still possible to solve the coupled system explicitly. For mean-field theory is reliable, and again we can solve the mean-field equations using the discretized approach. Finally, the discretized approach makes sense even in higher-dimensional spacetime. Although analytic solutions are still missing in the higher-dimensional case, numerical studies reveal an interesting structure and allow the identification of a fixed point where we can hope to define a genuine non-perturbative theory of four-dimensional quantum gravity.

Two-dimensional quantum gravity is identified as a second-class system which we convert into a first-class system via the Batalin-Fradkin (BF) procedure. Using the extended phase space method, we then formulate the theory in most general class of gauges. The conformal gauge action suggested by David, Distler and Kawai is derived from a first principle. We find a local, light-cone gauge action whose Becchi-Rouet-Stora-Tyutin invariance implies Polyakov's curvature equation $\partial_{-}R=\partial_{-}^{3}g_{++}=0$, revealing the origin of the $SL(2,R)$ Kac-Moody symmetry. The BF degree of freedom turns out be dynamically active as the Liouville mode in the conformal gauge, while in the light-cone gauge the conformal degree of freedom plays that r{\^o}le. The inclusion of the cosmological constant term in both gauges and the harmonic gauge-fixing are also considered.

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Quantum Ricci curvature has been introduced recently as a new, geometric observable characterizing the curvature properties of metric spaces, without the need for a smooth structure. Besides coordinate invariance, its key features are scalability, computability and robustness. We demonstrate that these properties continue to hold in the context of nonperturbative quantum gravity, by evaluating the quantum Ricci curvature numerically in two-dimensional Euclidean quantum gravity, defined in terms of dynamical triangulations. Despite the well-known, highly nonclassical properties of the underlying quantum geometry, its Ricci curvature can be matched well to that of a five-dimensional round sphere.

Shaken, but not stirred—Potts model coupled to quantum gravity

The ϵ-expansion of two-dimensional quantum gravity

String vacuum backgrounds with covariantly constant null Killing vector and two-dimensional quantum gravity

We discuss two examples of infinite random graphs obtained as limits of finite statistical mechanical systems: a model of two-dimensional dis-cretized quantum gravity defined in terms of causal triangulated surfaces, and the Ising model on generic random trees. For the former model we describe a relation to the so-called uniform infinite tree and results on the Hausdorff and spectral dimension of two-dimensional space-time obtained in B. Durhuus, T. Jonsson, J.F. Wheater, J. Stat. Phys. 139, 859 (2010) are briefly outlined. For the latter we discuss results on the absence of spontaneous magnetization and argue that, in the generic case, the values of the Hausdorff and spectral dimension of the underlying infinite trees are not influenced by the coupling to an Ising model in a constant magnetic field (B. Durhuus, G.M. Napolitano, in preparation)

The derivation of the critical dimensions D=26 of the bosonic string theory based on the two-dimensional quantum gravity in the harmonic gauge is criticized. The conformal anomaly calculated in lowest-order approximation crucially depends on the presence of a certain part of the FP-ghost Lagrangian density. However, this part can be eliminated by field redefinition and, moreover, reduces to a total divergence in lowest-order approximation. Thus the assertion that the anomaly is proportional to (D−26) is groundless.

Gravity coupled to a scalar field in two dimensions is analyzed in the conformal gauge. We obtain the solution of a scalar field. The residual symmetry associated with the scalar field is found from the invariance of the action and the covariance of the equation of motion of a scalar field. This symmetry is naturally related to the fact that the energy-momentum tensor satisfies the transformation rule of the usual second-rank tensor under $\mathrm{SL}(2,R)\ensuremath{\bigotimes}\mathrm{SL}(2,R)$ group transformations for the coordinates ${x}^{+}$ and ${x}^{\ensuremath{-}}$.