P. Di Francesco

2D gravity and random matrices

We review recent progress in 2D gravity coupled to d < 1 conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related O(n) matrix models. For d < 1 matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising-like matter, the sum over topologies is reduced to the solution of nonlinear differential equations (the Painleve equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces.

SU(N) lattice integrable models associated with graphs

Abstract We explore the construction of RSOS critical integrable models attached to a graph, trying to extend Pasquiers construction from SU(2) to SU( N ), with main emphasis on the case of SU(3): the heights are the nodes of a graph, which encodes the allowed configurations. A class of graphs that are natural candidates for this construction is defined. In the case N = 3, they all seem to be related to finite subgroups of SU(3). For any N , they are associated with arbitrary representations of the SU (N) fusion algebra over matrices of non-negative integers. It is argued that these graphs should support a representation of the Hecke algebra.

Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models

Partition functions of critical 2D models on a torus can be derived from their microscopic formulation and their free field representation in the continuum limit. This is worked out explicitly for theO(n) andQ-state Potts model. Forn orQ integer we recover results obtained from conformal invariance, but our procedure also extends to nonintegral values. In the latter case the expansion on characters of the Virasoro algebra involves real coefficients of either sign. The operator content of both models is discussed in detail.

Geodesic distance in planar graphs

We derive the exact generating function for planar maps (genus zero fatgraphs) with vertices of arbitrary even valence and with two marked points at a fixed geodesic distance. This is done in a purely combinatorial way based on a bijection with decorated trees, leading to a recursion relation on the geodesic distance. The latter is solved exactly in terms of discrete soliton-like expressions, suggesting an underlying integrable structure. We extract from this solution the fractal dimensions at the various (multi)-critical points, as well as the precise scaling forms of the continuum two-point functions and the probability distributions for the geodesic distance in (multi)-critical random surfaces. The two-point functions are shown to obey differential equations involving the residues of the KdV hierarchy.

Quantum intersection rings

Within the broadly defined subject of topological field theory E. Witten suggested in [1] to study generalized “intersection numbers” on a compactified moduli space \({\bar M_{g,n}}\) of Riemann surfaces. These are computed by integrating pullbacks of appropriate forms on a target Kahler manifold obtained through holomorphic maps of marked surfaces. The corresponding axioms, discussed by R. Dijkgraaf and E. and H. Verlinde [2], were investigated by M. Kontsevich and Y. Manin [3] and lead for a subclass of targets to surprising results on the enumeration of rational curves. Our purpose here is to study a few illustrative examples and to check some of them using the most primitive tools of geometry.

CRITICAL ISING CORRELATION FUNCTIONS IN THE PLANE AND ON THE TORUS

Abstract Explicit expressions are given for all correlation functions of spin, disorder and energy operators of the critical Ising model in the plane or on the torus. Formulae for insertions of energy-momentum tensors may also be written, giving access to correlation functions of secondary fields. Two alternative methods are used throughout the paper: bosonization of the fermion representation of the Ising model or use of the free field (orbifold) interpretation of the Ashkin-Teller model. Many identities result from the consistency between these alternative approaches.

Census of planar maps: from the one-matrix model solution to a combinatorial proof

We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an alternative and purely combinatorial solution to the problem of counting arbitrary planar maps with prescribed vertex degrees.

LAUGHLIN'S WAVE FUNCTIONS, COULOMB GASES AND EXPANSIONS OF THE DISCRIMINANT

In the context of the fractional quantum Hall effect, we investigate Laughlins ansatz for the ground state wave function at fractional filling of the lowest Landau level. Interpreting its normalization in terms of a one-component plasma, we find the effect of an additional quadrupolar field on the free energy, and derive estimates for the thermo dynamically equivalent spherical plasma. In the second part of the paper, we present various methods for expanding the wave function in terms of Slater determinants, and obtain sum rules for the coefficients. We also address the apparently simpler question of counting the number of such Slater states using the theory of integral polytopes.

GENERALIZED COULOMB-GAS FORMALISM FOR TWO DIMENSIONAL CRITICAL MODELS BASED ON SU(2) COSET CONSTRUCTION

Conformal theories associated with the coset construction SU(2)k × SU(2)JSU(2)k+t are described by a generalized Coulomb-gas formalism. An integrable lattice model obtained by k fusions of the 6-vertex model has a continuous critical line containing the level-k Wess-ZuminoWitten model. Its continuum limit is described by tensor products of the parafermionic and free bosonic sectors. Modification of its boundary conditions by the introduction of floating charges, yields the coset models. The discussion extends to non-unitary models. Special attention is paid to the case k = 2, corresponding to N = 1 supersymmetry.

Modular invariance in non-minimal two-dimensional conformal theories

Abstract We construct modular invariants in non-minimal conformal theories of central charge c c = 1 theories, and describe a free field with defect lines on a torus. This is applied to the determination of partition functions for critical Q -state Potts and O( n ) models, with a special emphasis on the polymer ( n = 0) case.