Free field primaries in general dimensions: Counting and construction with rings and modules

TL;DR

This paper introduces lowest weight polynomials (LWPs) linked to $so(d,2)$ representation theory, providing a new algebraic framework for counting and constructing primary fields in free scalar theories across dimensions.

Contribution

It establishes a novel algebraic and geometric approach to classify and construct primary fields using rings, modules, and quadratic algebras, revealing new mathematical structures.

Findings

LWPs correspond to primary fields in free scalar theories

A quotient of polynomial rings characterizes LWPs

A binomial identity underpins construction algorithms

Abstract

We define lowest weight polynomials (LWPs), motivated by $so(d,2)$ representation theory, as elements of the polynomial ring over $ d \times n $ variables obeying a system of first and second order partial differential equations. LWPs invariant under $S_n$ correspond to primary fields in free scalar field theory in $d$ dimensions, constructed from $n$ fields. The LWPs are in one-to-one correspondence with a quotient of the polynomial ring in $ d \times (n-1) $ variables by an ideal generated by $n$ quadratic polynomials. The implications of this description for the counting and construction of primary fields are described: an interesting binomial identity underlies one of the construction algorithms.The product on the ring of LWPs can be described as a commutative star product. The quadratic algebra of lowest weight polynomials has a dual quadratic algebra which is non-commutative. We discuss the possible physical implications of this dual algebra.