N = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries

• Published in: The journal of high energy physics 2023-02, Vol.2023 (2), p.176
• Main Authors: de Mello Koch, Robert, Ramgoolam, Sanjaye
• Format: Article
• Language: English
• Subjects: Algebra; Classical and Quantum Gravitation; Differential equations; Elementary Particles; Generators; High energy physics; Operators (mathematics); Physics; Physics and Astronomy; Polynomials; Quantum Field Theories; Quantum Field Theory; Quantum mechanics; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Representations; Rings (mathematics); String Theory; Theoretical physics
• ISSN: 1029-8479
• DOI: 10.1007/JHEP02(2023)176

A bstract The structure of half-BPS representations of psu(2 , 2 | 4) leads to the definition of a super-polynomial ring R (8 | 8) which admits a realisation of psu(2 , 2 | 4) in terms of differential operators on the super-ring. The character of the half-BPS fundamental field representation encodes the resolution of the representation in terms of an exact sequence of modules of R (8 | 8). The half-BPS representation is realized by quotienting the super-ring by a quadratic ideal, equivalently by setting to zero certain quadratic polynomials in the generators of the super-ring. This description of the half-BPS fundamental field irreducible representation of psu(2 , 2 | 4) in terms of a super-polynomial ring is an example of a more general construction of lowest-weight representations of Lie (super-) algebras using polynomial rings generated by a commuting subspace of the standard raising operators, corresponding to positive roots of the Lie (super-) algebra. We illustrate the construction using simple examples of representations of su(3) and su(4). These results lead to the definition of a notion of quantum mechanical emergence for oscillator realisations of symmetries, which is based on ideals in the ring of polynomials in the creation operators.