Beyond the Standard Model with Six-Dimensional Spinors
Six-dimensional spinors with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>3</mn...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Online Access: | https://www.mdpi.com/2571-712X/6/1/8 |
| _version_ | 1797609640972779520 |
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| author | David Chester Alessio Marrani Michael Rios |
| author_facet | David Chester Alessio Marrani Michael Rios |
| author_sort | David Chester |
| collection | DOAJ |
| description | Six-dimensional spinors with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> symmetry are utilized to efficiently encode three generations of matter. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mrow><mn>8</mn><mo>(</mo><mo>−</mo><mn>24</mn><mo>)</mo></mrow></msub></semantics></math></inline-formula> is shown to contain physically relevant subgroups with representations for GUT groups, spacetime symmetries, three generations of the standard model fermions, and Higgs bosons. Pati–Salam, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>5</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>10</mn><mo>)</mo></mrow></semantics></math></inline-formula> grand unified theories are found when a single generation is isolated. For spacetime symmetries, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> may be used for conformal symmetry, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>d</mi><msub><mi>S</mi><mn>5</mn></msub><mo>→</mo><mi>d</mi><msub><mi>S</mi><mn>4</mn></msub></mrow></semantics></math></inline-formula>, or simply broken to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> of a Minkowski space. Another class of representations finds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> and can give <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>d</mi><msub><mi>S</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> with various GUTs. An action for three generations of fermions in the Majorana–Weyl spinor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mn mathvariant="bold">128</mn></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow></semantics></math></inline-formula> is found with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> flavor symmetry inside <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mrow><mn>8</mn><mo>(</mo><mo>−</mo><mn>24</mn><mo>)</mo></mrow></msub></semantics></math></inline-formula>. The <b>128</b> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>12</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> can be regarded as the tangent space to a particular pseudo-Riemannian form of the octo-octonionic Rosenfeld projective plane <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mrow><mn>8</mn><mo>(</mo><mo>−</mo><mn>24</mn><mo>)</mo></mrow></msub><mo>/</mo><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mrow><mo>(</mo><mn>12</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mi mathvariant="double-struck">O</mi><mi>s</mi><mi>x</mi><mi mathvariant="double-struck">O</mi><mo>)</mo></mrow><msup><mi mathvariant="double-struck">P</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-11T06:03:16Z |
| format | Article |
| id | doaj.art-711ac0ae238e4ce5a725c231c9891847 |
| institution | Directory Open Access Journal |
| issn | 2571-712X |
| language | English |
| last_indexed | 2024-03-11T06:03:16Z |
| publishDate | 2023-01-01 |
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| spelling | doaj.art-711ac0ae238e4ce5a725c231c98918472023-11-17T13:08:35ZengMDPI AGParticles2571-712X2023-01-016114417210.3390/particles6010008Beyond the Standard Model with Six-Dimensional SpinorsDavid Chester0Alessio Marrani1Michael Rios2Quantum Gravity Research, Topanga, CA 90290, USAInstituto de Física Teorica, Universidad de Murcia, Campus de Espinardo, E-30100 Murcia, SpainQuantum Gravity Research, Topanga, CA 90290, USASix-dimensional spinors with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> symmetry are utilized to efficiently encode three generations of matter. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mrow><mn>8</mn><mo>(</mo><mo>−</mo><mn>24</mn><mo>)</mo></mrow></msub></semantics></math></inline-formula> is shown to contain physically relevant subgroups with representations for GUT groups, spacetime symmetries, three generations of the standard model fermions, and Higgs bosons. Pati–Salam, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>5</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>10</mn><mo>)</mo></mrow></semantics></math></inline-formula> grand unified theories are found when a single generation is isolated. For spacetime symmetries, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> may be used for conformal symmetry, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>d</mi><msub><mi>S</mi><mn>5</mn></msub><mo>→</mo><mi>d</mi><msub><mi>S</mi><mn>4</mn></msub></mrow></semantics></math></inline-formula>, or simply broken to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> of a Minkowski space. Another class of representations finds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> and can give <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>d</mi><msub><mi>S</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> with various GUTs. An action for three generations of fermions in the Majorana–Weyl spinor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mn mathvariant="bold">128</mn></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>4</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow></semantics></math></inline-formula> is found with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> flavor symmetry inside <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mrow><mn>8</mn><mo>(</mo><mo>−</mo><mn>24</mn><mo>)</mo></mrow></msub></semantics></math></inline-formula>. The <b>128</b> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>12</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> can be regarded as the tangent space to a particular pseudo-Riemannian form of the octo-octonionic Rosenfeld projective plane <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mrow><mn>8</mn><mo>(</mo><mo>−</mo><mn>24</mn><mo>)</mo></mrow></msub><mo>/</mo><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mrow><mo>(</mo><mn>12</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mi mathvariant="double-struck">O</mi><mi>s</mi><mi>x</mi><mi mathvariant="double-struck">O</mi><mo>)</mo></mrow><msup><mi mathvariant="double-struck">P</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2571-712X/6/1/8beyond the standard modelgraviGUT6D spinorsmodel buildingrepresentation theory |
| spellingShingle | David Chester Alessio Marrani Michael Rios Beyond the Standard Model with Six-Dimensional Spinors Particles beyond the standard model graviGUT 6D spinors model building representation theory |
| title | Beyond the Standard Model with Six-Dimensional Spinors |
| title_full | Beyond the Standard Model with Six-Dimensional Spinors |
| title_fullStr | Beyond the Standard Model with Six-Dimensional Spinors |
| title_full_unstemmed | Beyond the Standard Model with Six-Dimensional Spinors |
| title_short | Beyond the Standard Model with Six-Dimensional Spinors |
| title_sort | beyond the standard model with six dimensional spinors |
| topic | beyond the standard model graviGUT 6D spinors model building representation theory |
| url | https://www.mdpi.com/2571-712X/6/1/8 |
| work_keys_str_mv | AT davidchester beyondthestandardmodelwithsixdimensionalspinors AT alessiomarrani beyondthestandardmodelwithsixdimensionalspinors AT michaelrios beyondthestandardmodelwithsixdimensionalspinors |