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

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.

50 pages + Appendices; V2 : minor typos corrected

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• Permutation Invariant Gaussian Matrix Models
• Tensorial generalization of characters
• On generalized Macdonald polynomials
• Identifying Large Charge Operators
• $\mathcal{N}=4$ SYM, (super)-polynomial rings and emergent quantum mechanical symmetries
• Perturbative 4D conformal field theories and representation theory of diagram algebras