Algebraic-geometrical formulation of two-dimensional quantum gravity

We find a volume form on moduli space of double-punctured Riemann surfaces whose integral satisfies the Painlevé I recursion relations of the genus expansion of the specific heat of 2D gravity. This allows us to express the asymptotic expansion of the specific heat as an integral on an infinite-dimensional moduli space in the spirit of the Friedan-Shenker approach. We outline a conjectural derivation of such recursion relations using the Duistermaat-Heckman theorem.Partly supported by the European Community Research Programme Gauge theories, applied supersymmetry and quantum gravity, contract SC1-CT92-0789     

Generating functions in two-dimensional quantum gravity

We solve general 1-matrix models without taking the double scaling limit. A method of computing generating functions is presented. We calculate the generating functions for a simple and a double torus. Our method is also applicable to higher genus. Each generating function can be expressed by a “specific heat” function for the sphere. Universal terms, which survived in the double scaling limit can be easily picked out from our exact solutions. We also find that the regular part of the spherical generating function is at most bilinear in the coupling constants of the source terms.     

Infinite conformal symmetry in two-dimensional quantum field theory

We present an investigation of the massless, two-dimentional, interacting field theories. Their basic property is their invariance under an infinite-dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of Virasoro algebra, and that the correlation functions are built up of the “conformal blocks” which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the degenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the systems of linear differential equations.     

Operator product expansion in two-dimensional quantum gravity

We consider correlation functions of operators and the operator product expansion in two-dimensional quantum gravity. First we introduce correlation functions with geodesic distances between operators kept fixed. Next, by making two of the operators closer, we examine whether there exists an analog of the operator product expansion in ordinary field theories. Our results suggest that the operator product expansion holds in quantum gravity as well, though special care should be taken regarding the physical meaning of fixing geodesic distances on a fluctuating geometry.     

Vacuum structure in quantum gravity

The physical interpretation of the Eguchi-Hanson (EH) instanton solution is suggested. This solution describes the tunneling transition which connects vacua with opposite orientations. The diagonal vacuum states are symmetric and antisymmetric superpositions of states with a given orientation. The EH instantons lead to appearance of the induced Einstein term and of the large induced cosmological term in the effective action of the conformal gravity theory.     

Explicit solution forp-q duality in two-dimensional quantum gravity

By using Sato's infinite dimensional Grassmannian theory of the KP hierarchy, we study the global structure of the theory space of 2D quantum gravity coupled to various minimal conformal fields labeled by a pair of integers (p, q). After giving a rigorous proof of the equivalence of Douglas's equation and the Schwinger-Dyson equation (W-constrainton a function), we establish thep-q duality of the (p, q) quantum gravity at Green's function level. As an application, we discuss the metamorphosis of operators under unitarity-preserving renormalization group flows.Address after September 1, 1992: Newman Laboratory, Cornell University, Ithaca, NY 14853-5001, USA.     

Classical solutions of two-dimensional Grassmannian models

We present classical solutions of two-dimensional Euclidean Grassmannian σ models (harmonic maps) and discuss some of their properties.     

Geometry of a two-dimensional quantum gravity: Numerical study

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 à 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 ∫(φ)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.     

Time-dependent quantum scattering in 2+1 dimensional gravity

The propagation of a localized wave packet in the conical space-time created by a pointlike massive source in 2+1 dimensional gravity is analyzed. The scattering amplitude is determined and shown to be finite along the classical scattering directions due to interference between the scattered and the transmitted wave functions. The analogy with diffraction theory is emphasized.This work is supported in part by funds provided by the U. S. Department of Energy (D.O.E.) under cooperative agreement #DE-FC02-94ER40818.     

Unique effective action in two dimensional induced quantum gravity

The gauge dependence of the effective action in two dimensional induced gravity with non-local action is investigated. The unique effective action in this theory on flat background is also discussed. The configuration-space metric dependence of the unique effective action is studied.On leave from Tomsk Pedagogical Inst., Tomsk, Russia Address after 1.08.92: Dept. of Physics, Hiroshima Univ., Higashi-Hiroshima 724, Japan     

Infinite dimensional geometry and quantum field theory of strings. III. Infinite dimensional W-geometry of a second quantized free string

The present paper is devoted to various objects of the infinite dimensional W-geometry of a second quantized free string. Our purpose is to include the W-symmetries into the general infinite dimensional geometrical picture related to the quantum field theory of strings, which was described in part I of this series of papers. It is done by replacing the Lie algebra of all infinitesimal reparametrizations of a string world-sheet by the Lie quasi(pseudo)algebra of classical W-transformations (Gervais-Matsuo quasi(pseudo)algebra) as well as the Virasoro algebra by the central extended enlarged Gervais-Matsuo quasi(pseudo)algebra. A way to obtain W-algebras from the classical W-transformations (i.e. Gervais-Matsuo Lie quasi(pseudo)algebra) is proposed. A relationship between Gervais-Matsuo differential W-geometry and the Batalin-Weinstein-Karasev-Maslov approach to the geometry of nonlinear Poisson brackets as well as L.V. Sabinin's program of “nonlinear geometric algebra” is mentioned.     

Operator coproduct-realization of quantum group transformations in two-dimensional gravity I

A simple connection between the universalR matrix ofU _q(sl(2)) (for spins 1/2 andJ) and the required form of the coproduct action of the Hilbert space generators of the quantum group symmetry is put forward. This leads us to an explicit operator realization of the coproduct action on the covariant operators. It allows us to derive the expected quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of theU _q(sl(2)) algebra realized by (what we call) fixed point commutation relations. This is explained by showing on a general ground that the link between the algebra of field transformations and that of the coproduct generators is much weaker than previously thought. The central charges of our extendedU _q(sl(2)) algebra, which includes the Liouville zero-mode momentum in a non-trivial way, are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry related to the presence of both of the screening charges of 2D gravity.Partially supported by the EC contracts CHRXCT920069 and CHRXCT920035.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud.     

Unique effective action in two-dimensional quantum gravity with nonlocal action

The gauge dependence of the effective action in two-dimensional induced quantum gravity is investigated. A unique effective action is found, and its dependence on the metric of the configuration space is studied.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 97–100, January, 1992.