$\mathcal{N}=4$ SYM, (super)-polynomial rings and emergent quantum mechanical symmetries

Robert de Mello Koch; Sanjaye Ramgoolam

doi:10.1007/JHEP02(2023)176

paper

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

arXiv:2211.04271 · doi:10.1007/JHEP02(2023)176

Abstract

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.

arXiv comment60 pages, no figures; v2: Mathematica notebook added to submission

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

Abstract

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