Large-N Gauge Theories

Four pedagogical Lectures at the NATO-ASI on "Quantum Geometry" in Akureyri, Iceland, August 1999. Contents: 1. O(N) Vector Models, 2. Large-N QCD, 3. QCD in Loop Space, 4. Large-N ReductionComment: Lectures at the 1999 NATO-ASI on "Quantum Geometry" in Akureyri, Iceland; Latex, 69pp, 23 figure

QCD String as an Effective String

There are two cases where QCD string is described by an effective theory of long strings: the static potential and meson scattering amplitudes in the Regge regime. I show how the former can be solved in the mean-field approximation, justified by the large number of space-time dimensions, and argue that it turns out to be exact for the Nambu--Goto string. By adding extrinsic curvature I demonstrate how the tachyonic instability of the ground-state energy can be cured by operators less relevant in the infrared.Comment: Talk: 12pp., 3 fig

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

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

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

Implementation of the Duality between Wilson loops and Scattering Amplitudes in QCD

We generalize modern ideas about the duality between Wilson loops and scattering amplitudes in ${\cal N}$=4 SYM to large-N (or quenched) QCD. We show that the area-law behavior of asymptotically large Wilson loops is dual to the Regge-Veneziano behavior of scattering amplitudes at high energies and fixed momentum transfer, when quark mass is small and/or the number of particles is large. We elaborate on this duality for string theory in a flat space, identifying the asymptotes of the disk amplitude and the Wilson loop of large-N QCD.Comment: REVTex, 6 pages, 1 figure; v3: refs added; v4pp. to appear in PR

Exact Multiparticle Amplitudes at Threshold in ϕ4\phi^4 Theories with Softly Broken O(∞)O(\infty) Symmetry

We consider the problem of multiparticle production at threshold in a $\phi^4$-theory with an $O(N_1$$+$$N_2)$ symmetry softly broken down to $O(N_1)\times O(N_2)$ by nonequal masses. We derive the set of recurrence relations between the multiparticle amplitudes which sums all relevant diagrams with arbitrary number of loops in the large-$N$ limit with fixed number of produced particles. We transform it into a quantum mechanical problem and show how it can be obtained directly from the operator equations of motion by applying the factorization at large $N$. We find the exact solutions to the problem by using the Gelfand--Diki\u{\i} representation of the diagonal resolvent of the Schr\"{o}dinger operator. The result coincides with the tree amplitudes while the effect of loops is the renormalization of the coupling constant and masses. The form of the solution is due to the fact that the exact amplitude of the process $2$\ra$n$ vanishes at $n$$>$$2$ on mass shell when averaged over the $O(N_{1,2})$-indices of incoming particles. We discuss what dynamical symmetry is behind this property. We also give an exact solution in the large-$N$ limit for the model of the $O(N)$$+$$singlet$ scalar particle with the spontaneous breaking of a reflection symmetry.Comment: Latex, 33 pages, NBI-HE-94-3