N $$ \mathcal{N} $$ = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries
Abstract The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polynomial ring R $$ \mathcal{R} $$ (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 representatio...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-02-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP02(2023)176 |
| _version_ | 1797779121181294592 |
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| author | Robert de Mello Koch Sanjaye Ramgoolam |
| author_facet | Robert de Mello Koch Sanjaye Ramgoolam |
| author_sort | Robert de Mello Koch |
| collection | DOAJ |
| description | Abstract The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polynomial ring R $$ \mathcal{R} $$ (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 $$ \mathcal{R} $$ (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. |
| first_indexed | 2024-03-12T23:26:11Z |
| format | Article |
| id | doaj.art-68ad63ceb3054d8c89423cf1c80decd2 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-03-12T23:26:11Z |
| publishDate | 2023-02-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-68ad63ceb3054d8c89423cf1c80decd22023-07-16T11:06:29ZengSpringerOpenJournal of High Energy Physics1029-84792023-02-012023215110.1007/JHEP02(2023)176N $$ \mathcal{N} $$ = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetriesRobert de Mello Koch0Sanjaye Ramgoolam1School of Science, Huzhou UniversityCentre for Theoretical Physics, School of Physical and Chemical Sciences, Queen Mary University of LondonAbstract The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polynomial ring R $$ \mathcal{R} $$ (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 $$ \mathcal{R} $$ (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.https://doi.org/10.1007/JHEP02(2023)176AdS-CFT CorrespondenceSupersymmetric Gauge TheoryDifferential and Algebraic GeometryGlobal Symmetries |
| spellingShingle | Robert de Mello Koch Sanjaye Ramgoolam N $$ \mathcal{N} $$ = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries Journal of High Energy Physics AdS-CFT Correspondence Supersymmetric Gauge Theory Differential and Algebraic Geometry Global Symmetries |
| title | N $$ \mathcal{N} $$ = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries |
| title_full | N $$ \mathcal{N} $$ = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries |
| title_fullStr | N $$ \mathcal{N} $$ = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries |
| title_full_unstemmed | N $$ \mathcal{N} $$ = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries |
| title_short | N $$ \mathcal{N} $$ = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries |
| title_sort | n mathcal n 4 sym super polynomial rings and emergent quantum mechanical symmetries |
| topic | AdS-CFT Correspondence Supersymmetric Gauge Theory Differential and Algebraic Geometry Global Symmetries |
| url | https://doi.org/10.1007/JHEP02(2023)176 |
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