Gauge equivalence in two-dimensional gravity.

We show how to formulate a lattice gauge theory whose naive continuum limit corresponds to two-dimensional (Euclidean) quantum gravity including a positive cosmological constant. More precisely the resultant continuum theory corresponds to gravity in a first-order formalism in which the local frame and spin connection are treated as independent fields. Recasting this lattice theory as a tensor network allows us to study the theory at strong coupling without encountering a sign problem. In two dimensions this tensor network is exactly soluble and we show that the system has a series of critical points that occur for pure imaginary coupling and are associated with first order phase transitions. We then augment the action with a Yang-Mills term which allows us to control the lattice spacing and show how to apply the tensor renormalization group to compute the free energy and look for critical behavior. Finally we perform an analytic continuation in the gravity coupling in this extended model and show that its critical behavior in a certain scaling limit depends only on the topology of the underlying lattice. We also show how the lattice gauge theory can be naturally generalized to generate the Polyakov or Liouville action for two dimensional quantum gravity.

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 ${x}^{\ensuremath{-}}$.

The effect of the 2D gravity on the β-functions of a non-linear σ-model

Operator product expansion in two-dimensional quantum gravity

Light-cone gauge formulation of fields in AdS space and conformal field theory in flat space adapted for the study of AdS/CFT correspondence is developed. Arbitrary spin mixed-symmetry fields in AdS space and arbitrary spin mixed-symmetry currents, shadows, and conformal fields in flat space are considered on an equal footing. For the massless and massive fields in AdS and the conformal fields in flat space, simple light-cone gauge actions leading to decoupled equations of motion are found. For the currents and shadows, simple expressions for all 2-point functions are also found. We demonstrate that representation of conformal algebra generators on space of currents, shadows, and conformal fields can be built in terms of spin operators entering the light-cone gauge formulation of AdS fields. This considerably simplifies the study of AdS/CFT correspondence. Light-cone gauge actions for totally symmetric arbitrary spin long conformal fields in flat space are presented. We apply our approach to the study of totally antisymmetric (one-column) and mixed-symmetry (two-column) fields in AdS space and currents, shadows, and conformal fields in flat space.

Lattice gravity and strings

Path integral over conformally self-dual geometries

The ϵ-expansion of 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.

Comments on the importance of being over-constrained

We show that the light-cone and conformal gauges are equivalent in the sense that the well known SL (2, ℝ) current structure in the light-cone gauge is obtained from the results in the conformal one under a proper coordinate transformation. This result is due to the residual symmetry of the scalar field Φ.

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.

Quantization of the pure $1+1$ dimensional dilaton gravity is examined in the light-cone gauge. It is found that the total action including ghosts generates a $c=0$ free conformal field theory without modification of the classical action, which is required in the conformal gauge. We also study semiclassical equations of the dilaton gravity coupled to $N$ scalar fields. It is shown that the black hole singularity is not removed even for $N<24$ in the light-cone gauge. This indicates that the semiclassical analysis breaks down for small $N$.

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.

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