Infinite Dimensional Free Algebra and the Forms of the Master Field

TL;DR

This paper introduces an infinite dimensional free algebra relevant to large N SU(N) theories, providing new algebraic forms of the master field and novel free-algebraic representations of the planar Schwinger-Dyson equations.

Contribution

It constructs a new free algebra framework at large N, generalizes Chebyshev polynomials, and derives new algebraic forms of the master field and Schwinger-Dyson equations.

Findings

Established an infinite dimensional free algebra at large N

Derived algebraic forms of the master field including Gopakumar-Gross form

Provided new free-algebraic representations of planar Schwinger-Dyson equations

Abstract

We find an infinite dimensional free algebra which lives at large N in any SU(N)-invariant action or Hamiltonian theory of bosonic matrices. The natural basis of this algebra is a free-algebraic generalization of Chebyshev polynomials and the dual basis is closely related to the planar connected parts. This leads to a number of free-algebraic forms of the master field including an algebraic derivation of the Gopakumar-Gross form. For action theories, these forms of the master field immediately give a number of new free-algebraic packagings of the planar Schwinger-Dyson equations.