W_\infty and w_\infty Gauge Theories and Contraction

TL;DR

This paper develops a method to construct Winf and winf gauge theories using local fields in higher dimensions, explores their classical contraction, and relates quantum Winf theories to large N limits of SU(N) gauge theories.

Contribution

It introduces a new construction framework for Winf and winf gauge theories and clarifies their relationship to SU(N) gauge theories through contraction and quantum analysis.

Findings

Winf gauge theory is the large N limit of SU(N) gauge theory with proper renormalization.

winf gauge theory cannot be obtained as a large N limit.

A classical contraction procedure relates Winf and winf gauge theories.

Abstract

We present a general method of constructing Winf and winf gauge theories in terms of d+2 dimensional local fields. In this formulation the \Winf gauge theory Lagrangians involve non-local interactions, but the winf theories are entirely local. We discuss the so-called classical contraction procedure by which we derive the Lagrangian of winf gauge theory from that of the corresponding Winf gauge theory. In order to discuss the relationship between quantum Winf and quantum winf gauge theory we solve d=1 gauge theory models of a Higgs field exactly by using the collective field method. Based on this we conclude that the Winf gauge theory can be regarded as the large N limit of the corresponding SU(N) gauge theory once an appropriate coupling constant renormalization is made, while the winf gauge theory cannot be.