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

Understand

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.