3D Gravity Modelling Research Articles

Conformal matrix models as an alternative to conventional multi-matrix models

We introduce conformal multi-matrix models (CMMs) as an alternative to conventional multi-matrix model description of two-dimensional gravity interacting with c < 1 matter. We define CMMs as solutions to (discrete) extended Virasoro constraints. We argue that the so defined alternatives of multi-matrix models represent the same universality classes in the continuum limit, while at the discrete level they provide explicit solutions to the multi-component KP-hierarchy and by definition satisfy the discrete W-constraints. We prove that discrete CMMs coincide with the ( p, q)-series of 2d gravity models in a well-defined continuum limit, thus demonstrating that they provided a proper generalization of the hermitian one-matrix model.

Large order behaviour of 2D gravity coupled to D&lt;1 matter

We discuss the large order behaviour and Borel summability of the topological expansion of models of 2D gravity coupled to general $(p,q)$ conformal matter. In a previous work it was proven that at large order $k$ the string susceptibility had a generic $a^k\Gamma(2k-\ud)$ behaviour. Moreover the constant $a$, relevant for the problem of Borel summability, was determined for all one-matrix models. We here obtain a set of equations for this constant in the general $(p,q)$ model. String equations can be derived from the construction of two differential operators $P,Q$ satisfying canonical commutation relations $[P,Q]=1$. We show that the equation for $a$ is determined by the form of the operators $P,Q$ in the spherical or semiclassical limits. The results for the general one-matrix models are then easily recovered. Moreover, since for the $(p,q)$ string models such $p=(2m+1)q\pm1$ the semiclassical forms of $P,Q$ are explicitly known, the large order behaviour is completely determined. This class contains all unitary $(q+1,q)$ models for which the answer is specially simple. As expected we find that the topological expansion for unitary models is not Borel summable. \preprint{SPhT/92-163}, Plain-TeX, macro harvmac

The quantum group structure of 2D gravity and minimal models II: The genus-zero chiral bootstrap

The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants whose derivation is one aim of the present work. Our explicit formulae are new, to our knowledge, in spite of the numerous studies of this problem. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the author (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with Liouville momenta going to $\infty$ in a suitable way. The introduction of quantum-group-covariant operators in the three dimensional picture gives a generalisation of the quantum-group version of discrete three-dimensional gravity that includes tetrahedra associated with 3-j symbols and universal R-matrix elements. Altogether the present work gives the concrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models.

Scalar-tensor quantum gravity in two dimensions

We discuss some classical and quantum properties of 2d gravity models involving metric and a scalar field. Different models are parametrized in terms of a scalar potential. We show that a general Liouville-type model with exponential potential and linear curvature coupling is renormalisable at the quantum level while a particular model (corresponding to D = 2 graviton-dilaton string effective action and having a back-hole solution) is finite. We use the condition of a “split” Weyl symmetry to suggest possible expressions for the “effective action” which includes the quantum anomaly term.

Matrix models with γ string>0

Within the framework of c=1 matrix models, we consider multi-matrix models, i.e. the quantum mechanics of multi-matrix models. A connection is established between a D-matrix model and a D-dimensional gas of fermions (bosons) for odd (even) values of D. A statistical mechanical analysis yields the scaling law for the free energy, and hence the susceptibility exponents for the various models. The exponents turn out to be positive for the multi-matrix models, suggesting that these could represent models of 2D gravity coupled to c>1 matter. However, a lower-bound on the mass-gap exponents is found (i.e. an upper bound on the Hausdroff dimension) which may render this identification unlikely. Nevertheless, we find certain qualitative features which would be expected of a c>1 theory. For instance, in addition to the positive susceptibility exponent, we find that whereas in the c=1 case the density of states itself diverges as one approaches the critical point, in the D-matrix models various derivatives of the density of states diverge, with the order of the derivative depending on D. This qualitatively different behaviour of the density of states could be a signal of the conjectured “phase transition” at c=1.

Unification of all string models with c

A one-matrix model is proposed, which nicely interpolates between double-scaling continuum limits of all multimatrix models. The interpolating partition function is always a KP τ-function and always obeys the L −1- constraint and the string equation. Therefore this model can be considered as a natural unification of all models of 2D-gravity (string models) with c⩽1.

The quantum group structure of 2D gravity and minimal models I

On the unit circle, an infinite family of chiral operators is constructed, whose exchange algebra is given by the universalR-matrix of the quantum groupSL(2) q . This establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group. The method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofSL(2) q . The formulae so derived, which generalize an earlier particular case worked out by Babelon, are remarkably compact and may be entirely written in terms of “q-deformed” factorials and binomial coefficients.

O( N) vector field theories in the double scaling limit

With the hope of getting some new insight into the problem arising in matrix models of 2D gravity when the dimension of matter is larger than 1, and inspired by several articles showing that O( N) invariant vector models have, at least in zero dimensions, nontrivial scaling limits, we study the large N behaviour of field theory vector models in d dimensions and discuss the nature the form of the critical behaviour. We show that there exists a doubble scaling limit where a bound state of the vector φ associated with the φ 2 composite operator becomes massless, while the fundamental field φ itself remains massive. The model can be described by an effective local interaction for the corresponding O( N) invariant field.

Large order behavior of non-perturbative gravity

We study the large order behavior and Borel summability of the topological expansion of models of 2D gravity coupled to matter. We treat in particular the cases of pure gravity, and gravity coupled to Yang-Lee, Ising, and tricritical Ising matter. We then discuss arbitrary one-matrix models and gravity coupled to general ( p, q) conformal matter, in particular showing that the genus- k contribution to their partition functions goes asymptotically for large k as Γ (2k− 1 2 ) . Finally, we show that the large order behavior analysis for the one-matrix models coincides with an instanton calculation of barrier penetration effects.

Integrability in random matrix models

We prove the equivalence between the recent matrix model formulation of 2D gravity and lattice integrable models. For even potentials this system is the Volterra hierarchy, and many properties of the continuum matrix model like the Virasoro conditions on the partition function stem directly from the integrability properties of the lattice model and its hamiltonian properties.

Instability of even m multicritical matrix models of 2D gravity

The hermitian one-matrix model of two-dimensional gravity is studied in the large N limit by the saddle-point method for a general even matrix potential giving a well defined partition function. A simple demonstration of the connection between instability of m even models and the existence of multi-arc phases is given.

Geometry of the string equations

The string equations of hermitian and unitary matrix models of 2D gravity are flatness conditions. These flatness conditions may be interpreted as the consistency conditions for isomonodromic deformation of an equation with an irregular singularity. In particular, the partition function of the matrix model is shown to be the tau function for isomonodromic deformation. The physical parameters defining the string equation are interpreted as moduli of meromorphic gauge fields, and the compatibility conditions can be interpreted as defining a “quantum” analog of a Riemann surface. In the latter interpretation, the equations may be viewed as compatibility conditions for transport on “quantum moduli space” of correlation functions in a theory of free fermions. We discuss how the free fermion field theory may be deduced directly from the matrix model integral. As an application of our formalism we discuss some properties of the BMP solutions of the string equations. We also mention briefly a possible connection to twistor theory.

The quantum group structure of 2D gravity and minimal models I

On the unit circle, an infinite family of chiral operators is constructed, whose exchange algebra is given by the universalR-matrix of the quantum groupSL(2) q . This establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group. The method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofSL(2) q . The formulae so derived, which generalize an earlier particular case worked out by Babelon, are remarkably compact and may be entirely written in terms of “q-deformed” factorials and binomial coefficients.

BRS analysis of 2D gravity models

Using the original BRS approach we determine, from the requirement of diffeomorphism invariance, the DDK action of 2D quantum gravity in conformal gauge. The necessity to take into account quantum corrections due to Liouville modes is pointed out. A quantum effective action for a pure 2D gravity with the classical action ∫d 2 x√− g NR is also determined.

Matrix Models of 2D Gravity and Isomonodromic Deformation

One of the principal goals of the theory of 2D gravity is making sense of the formal expression $$Z\left( {\mu ,\kappa ;{t_i}} \right) = \sum\limits_h {{{\int_{ME{T_h}} {dge} }^{\mu \int {\sqrt g + k\int {\sqrt g } + k} }}} {Z_{QFT\left( {{t_i}} \right)}}\left[ g \right]$$ (1.1) where we integrate over metrics g on surfaces with h handles with a weight defined by the Einstein-Hilbert action (μ is the cosmological constant and κ is Newton’s constant, or, equivalently, the string coupling) together with the partition function of some 2D quantum field theory, QFT(t i ). The parameters t i are coordinates on a subspace of the space of 2D field theories, or, equivalently, coordinates for a space of string backgrounds.