Collective-field Approach Research Articles

Solutions of coupled BPS equations for two-family Calogero and matrix models

We consider a large $N$, two-family Calogero and matrix model in the Hamiltonian, collective-field approach. The Bogomol'nyi limit appears and the solutions to the coupled Bogomol'nyi-Prasad-Sommerfeld equations are given by the static soliton configurations. We find all solutions close to constant and construct exact one-parameter solutions in the strong-weak dual case. Full classification of these solutions is presented.

Solitons and giants in matrix models

We present a method for solving BPS equations obtained in the collective-field approach to matrix models. The method enables us to find BPS solutions and quantum excitations around these solutions in the one-matrix model, and in general for the Calogero model. These semiclassical solutions correspond to giant gravitons described by matrix models obtained in the framework of AdS/CFT correspondence. The two-field model, associated with two types of giant gravitons, is investigated. In this duality-based matrix model we find the finite form of the $n$-soliton solution. The singular limit of this solution is examined and a realization of open-closed string duality is proposed.

2D Calogero model in the collective-field approach

We consider the large- N Calogero–Marchioro model in two dimensions in the Hamiltonian collective-field approach based on the 1 / N expansion. The Bogomol'nyi limit appears in the presence of the harmonic confinement. We investigate density fluctuations around the semiclassical uniform solution. The excitation spectrum splits into two branches depending on the value of the coupling constant. The ground state exhibits long-range order.

A COHOMOLOGICAL INTERPRETATION OF THE MIGDAL–MAKEENKO EQUATIONS

The equations of motion of quantum Yang–Mills theory (in the planar "large-N" limit), when formulated in loop-space are shown to have an anomalous term, which makes them analogous to the equations of motion of WZW models. The anomaly is the Jacobian of the change of variables from the usual ones, i.e. the connection one-form A, to the holonomy U. An infinite-dimensional Lie algebra related to this change of variables (the Lie algebra of loop substitutions) is developed, and the anomaly is interpreted as an element of the first cohomology of this Lie algebra. The Migdal–Makeenko equations are shown to be the condition for the invariance of the Yang–Mills generating functional Z under the action of the generators of this Lie algebra. Connections of this formalism to the collective field approach of Jevicki and Sakita are also discussed.

Quantum fluctuations of the Chern-Simons theory and dynamical dimensional reduction

We consider a large-N Chern-Simons theory for the attractive bosonic matter (Jackiw-Pi model) in the Hamiltonian collective-field approach based on the 1/N expansion. We show that the dynamics of low-lying density excitations around the ground-state vortex configuration is equivalent to that of the Sutherland model. The relationship between the Chern-Simons coupling constant lambda and the Calogero-Sutherland statistical parameter lambda_s signalizes some sort of statistical transmutation accompanying the dimensional reduction of the initial problem.

Multi-vortex solution in the Sutherland model

We consider the large-$N$ Sutherland model in the Hamiltonian collective-field approach based on the $1/N$ expansion. The Bogomol'nyi limit appears and the corresponding solutions are given by static-soliton configurations. They exist only for $l<1$, i.e. for the negative coupling constant of the Sutherland interaction. We determine their creation energies and show that they are unaffected by higher-order corrections. For $l=1$, the Sutherland model reduces to the free one-plaquette Kogut-Susskind model.

Vortices of the Chern-Simons theory in the collective field approach.

We formulate a collective field approach to a planar nonrelativistic system of bosons interacting via the Chern-Simons interaction. Adding the attractive $\ensuremath{\delta}$ potential this system can be solved and the vortex soliton appears. The ground state of this system is scale and inversion invariant. In the harmonic well this system is not stable against the creation of vortices. Leading and next to leading terms in the $\frac{1}{N}$ expansion of the ground-state energy are calculated.

Hadronization of quark flavour dynamics including three-quark forces

We investigate an effective chiral quark model containing besides a two-quark Nambu-Jona-Lasinio force also three-quark forces. By means of path-integral techniques this model is, at first, converted into a quark-diquark-meson-baryon theory and afterwards hadronized into a meson-baryon theory. This way, we have performed an approximative step-by-step reduction of a three-body problem to a two-body problem in the collective field approach.

COLLECTIVE-FIELD APPROACH TO ANYONIC SYSTEMS

We consider a large-N anyonic system in the Hamiltonian, collective-field approach. We derive a collective-field representation of the Hamiltonian. In some interesting cases we determine the ground-state energy and the corresponding anyon collective motions.

Loop space Hamiltonians and numerical methods for large- N gauge theories

We consider large- N gauge theories in the hamiltonian, collective field approach. We derive an alternative collective representation which leads to significant reduction when translation invariance is invoked. It allows for a simplified computer simulation of loop rearrangements and the development of numerical techniques in the hamiltonian, loop space formalism. We proceed to give numerical evidence for validity of our representation and outline a general numerical approach for solving large- N QCD in terms of gauge-invariant Wilson loop variables.

Collective field in Hamiltonian and Lagrangian (path-integral) approach: An application to O(N) Heisenberg spin operator

Collective field approach to O(N) Heisenberg spin system is discussed. Hamiltonian formulation is reviewed and connection with largeN limit is shown. Collective field is then introduced in the Lagrangian path-integral formulation and 1/N corrections to various quantities like mass-gap, beta-function are computed.

Collective field approach to the large- N limit: Euclidean field theories

We apply the collective field method to study the large-N behavior of euclidean field theories. We show how this method leads to an effective action whose stationary points determine the N → ∞ limit. This approach applies to Yang-Mills gauge theories and the effective action is given in terms of gauge-invariant Wilson loop variables.