Barriers in Quantum Gravity

I discuss recent progress in our understanding of two barriers in quantum gravity: $c > 1$ in the case of 2d quantum gravity and $D > 2$ in the case of Euclidean Einstein-Hilbert gravity formulated in space-time dimensions $D >2$.Comment: standard latex, 10 pages. (one year old contribution to Trieste workshop, but continued demand for preprints has motivated me to put it on the bulletin board), NBI-HE-93-3

Electroweak Magnetism, W-condensation and Anti-Screening

We review how external magnetic fields act as perfect probes of the non-abelian nature of the electroweak theory.Comment: Latex 12p. NBI-HE-93-1

Scaling behavior of regularized bosonic strings

We implement a proper-time UV regularisation of the Nambu-Goto string, introducing an independent metric tensor and the corresponding Lagrange multiplier, and treating them in the mean-field approximation justified for long strings and/or when the dimensions of space-time is large. We compute the regularised determinant of the 2d Laplacian for the closed string winding around a compact dimension, obtaining in this way the effective action, whose minimisation determines the energy of the string ground state in the mean-field approximation. We discuss the existence of two scaling limits when the cutoff is taken to infinity. One scaling limit reproduces the results obtained by the hypercubic regularisation of the Nambu-Goto string as well as by the use of the dynamical triangulation regularisation of the Polyakov string. The other scaling limit reproduces the results obtained by canonical quantisation of the Nambu-Goto string.Comment: 35 page

Multi-point functions of weighted cubic maps

We study the geodesic two- and three-point functions of random weighted cubic maps, which are obtained by assigning random edge lengths to random cubic planar maps. Explicit expressions are obtained by taking limits of recently established bivariate multi-point functions of general planar maps. We give an alternative interpretation of the two-point function in terms of an Eden model exploration process on a random planar triangulation. Finally, the scaling limits of the multi-point functions are studied, showing in particular that the two- and three-point functions of the Brownian map are recovered as the number of faces is taken to infinity.Comment: 28 pages, 7 figures, several details and clarifications adde

Geodesic distances in Liouville quantum gravity

In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice. This geodesic distance is used to numerically determine the Hausdorff dimension associated to shortest cycles of 2d quantum gravity on the torus coupled to conformal matter fields, showing agreement with a conjectured formula by Y. Watabiki. Finally, the numerical tools are put to test by quantitatively comparing the distribution of lengths of shortest cycles to the corresponding distribution in large random triangulations.Comment: 21 pages, 8 figure

String theory as a Lilliputian world

Lattice regularizations of the bosonic string allow no tachyons. This has often been viewed as the reason why these theories have never managed to make any contact to standard continuum string theories when the dimension of spacetime is larger than two. We study the continuum string theory in large spacetime dimensions where simple mean field theory is reliable. By keeping carefully the cutoff we show that precisely the existence of a tachyon makes it possible to take a scaling limit which reproduces the lattice-string results. We compare this scaling limit with another scaling limit which reproduces standard continuum-string results. If the people working with lattice regularizations of string theories are akin to Gulliver they will view the standard string-world as a Lilliputian world no larger than a few lattice spacings.Comment: 11 pages, 1 figur

2D Quantum Gravity Coupled to Renormalizable Matter Fields

We consider two-dimensional quantum gravity coupled to matter fields which are renormalizable, but not conformal invariant. Questions concerning the \b function and the effective action are addressed, and the effective action and the dressed renormalization group equations are determined for various matter potentials.Comment: 23 pages, Latex, NBI-HE-93-6

The matrix model for hypergeometric Hurwitz numbers

We present the multi-matrix models that are the generating functions for branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, $z_1$ and $z_n$. We take a sum over all possible ramifications at other $n-2$ points with the fixed length of the profile at $z_2$ and with the fixed total length of profiles at the remaining $n-3$ points. All these models belong to a class of hypergeometric Hurwitz models thus being tau functions of the Kadomtsev--Petviashvili (KP) hierarchy. In the case described above, we can present the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type \tr M_iM_{i+1}^{-1}. We describe the technique for evaluating spectral curves of such models, which opens the possibility of applying the topological recursion for developing $1/N^2$-expansions of these model. These spectral curves turn out to be of an algebraic type.Comment: 12 pages, 2 figures in LaTeX, contribution to the volume of TMPh celebrating the 75th birthday of A A Slavno

Semi-classical Dynamical Triangulations

For non-critical string theory the partition function reduces to an integral over moduli space after integrating over matter fields. The moduli integrand is known analytically for genus one surfaces. The formalism of dynamical triangulations provides us with a regularization of non-critical string theory and we show that even for very small triangulations it reproduces very well the continuum integrand when the central charge $c$ of the matter fields is large negative, thus providing a striking example of how the quantum fluctuations of geometry disappear when $c \to -\infty$.Comment: 11 pages, 5 figure