The First Thirty Years of Large-N Gauge Theory

I review some developments in the large-N gauge theory since 1974. The main attention is payed to: multicolor QCD, matrix models, loop equations, reduced models, 2D quantum gravity, free random variables, noncommutative theories, AdS/CFT correspondence.Comment: 13.1pp., Latex, 2 figs; v2: 2 refs added. Talk at Large Nc QCD 200

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

Semiclassical Regge trajectories of noncritical string and large-N QCD

By properly treating the path integral over the boundary value of the Liouville field (associated with reparametrizations of the boundary contour) in open string theory, we derive consistent off-shell scattering amplitudes in d=26 dimensions. In d<26 we consider a recently proposed boundary ansatz which reproduces a semiclassical correction to the classical string (known as the Luscher term) and obtain in the semiclassical approximation a linear Regge trajectory with the intercept (d-2)/24. We associate it with the quark-antiquark Regge trajectory in large-N QCD and explain why it dominates over perturbative QCD when t > -few GeV^2.Comment: 20pp., 2 figures; v2: minor changes, to appear in JHE

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

What Quantum Strings can tell us about Quantum Gravity

I describe the recent progress in resolving two problems of nonperturbative bosonic string inherited from 1980's. Both the lattice and KPZ-DDK no-go theorems can be bypassed thanks to specific features of the theory with diffeomorphism invariance.Comment: 12 pages, 2 figures. Talk at the International Conference on Quantum Field Theory, High-Energy Physics and Cosmology, Dubna July 18--21, 2022; v2: reduced to 8 page

Exact solution of higher-derivative conformal theory and minimal models

I investigate the two-dimensional four-derivative conformal theory that emerges from the Nambu-Goto string after the path-integration over all fields but the metric tensor. Using the method of singular products which accounts for tremendous cancellations in perturbation theory, I show the (intelligent) one-loop approximation to give an exact solution. It is conveniently described through the minimal models where the central charge $c$ in the Kac spectrum depends on the parameters of the four-derivative action. The relation is nonlinear so the domain of physical parameters is mapped onto $c<1$ thus bypassing the KPZ barrier of the Liouville action.Comment: 11 pages. arXiv admin note: text overlap with arXiv:2307.06295; v2: polishe