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arXiv · 1806.01085

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

Robert de Mello Koch·Sanjaye Ramgoolam
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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.

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Robert de Mello Koch, Sanjaye Ramgoolam. 2018-06-04. Free field primaries in general dimensions: Counting and construction with rings and modules. https://doi.org/10.1007/jhep08(2018)088

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