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 × 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×(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.

Classifying and constraining local four photon and four graviton S-matrices

“…the subgroup of double transpositions. 30 The importance of this subgroup is that it leaves all the Mandelstam variables s, t and u invariant. Another feature of this subgroup is that it is normal.…”

“…(2.27) 30 We label an element of S4 by the image of (1234) under that element. The group Z2 × Z2 consists of the two listed generators together with the identity permutation (1234) and a fourth element (4321).…”

“…It follows that local 4 scalar S-matrices are in one to one correspondence with the equivalence classes L 4 described in this subsubsection. Note that a complete classification of scalar field primaries transforming in any representation and with arbitrary number of φs has been carried out using algebraic methods in [29,30].…”

Permutation invariant Gaussian matrix models

“…It is a useful fact that the Clebsch-Gordan coefficients for V H ⊗ V H → V H can be usefully written in terms of the C a,i describing V H as a subspace of the natural representation. This has recently played a role in the explicit description of a ring structure on primary fields of free scalar conformal field theory [14]. It would be interesting to explore the more general construction of explicit Clebsch-Gordan coefficients and projectors in the representation theory of S D in terms of the C a,i .…”

On generalized Macdonald polynomials

“…An essential feature is that the Gaussian averages of Macdonald polynomials are Macdonald dimensions: this is a basic property character = character (1.1) of matrix and tensor models, presumably related to their super integrability, see [6][7][8][9][10][11] and [12][13][14][15][16][17][18][19] for related references. The simplest and reference realization of this formula is the Gaussian Hermitean matrix model average…”