{"title":"费米子和玻色子代数和相关狄拉克算子中的单元豪对偶性","authors":"Guner Muarem","doi":"10.21468/scipostphysproc.14.038","DOIUrl":null,"url":null,"abstract":"<jats:p>In this paper we use the canonical complex structure <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{J}</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>𝕁</mml:mi></mml:math></jats:alternatives></jats:inline-formula> on <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{R}^{2n}</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msup><mml:mi>ℝ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></jats:alternatives></jats:inline-formula> to introduce a twist of the symplectic Dirac operator. This can be interpreted as the bosonic analogue of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these Dirac operators is isomorphic to the Lie algebra <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathfrak{su}(1,2)</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mrow><mml:mi>𝔰</mml:mi><mml:mi>𝔲</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo></mml:mrow></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> which leads to the Howe dual pair <jats:inline-formula><jats:alternatives><jats:tex-math>(U(n),\\mathfrak{su}(1,2))</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>𝔰</mml:mi><mml:mi>𝔲</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo></mml:mrow><mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>.</jats:p>","PeriodicalId":355998,"journal":{"name":"SciPost Physics Proceedings","volume":"32 5","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Unitary Howe dualities in fermionic and bosonic algebras and related Dirac operators\",\"authors\":\"Guner Muarem\",\"doi\":\"10.21468/scipostphysproc.14.038\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>In this paper we use the canonical complex structure <jats:inline-formula><jats:alternatives><jats:tex-math>\\\\mathbb{J}</jats:tex-math><mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><mml:mi>𝕁</mml:mi></mml:math></jats:alternatives></jats:inline-formula> on <jats:inline-formula><jats:alternatives><jats:tex-math>\\\\mathbb{R}^{2n}</jats:tex-math><mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><mml:msup><mml:mi>ℝ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></jats:alternatives></jats:inline-formula> to introduce a twist of the symplectic Dirac operator. This can be interpreted as the bosonic analogue of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these Dirac operators is isomorphic to the Lie algebra <jats:inline-formula><jats:alternatives><jats:tex-math>\\\\mathfrak{su}(1,2)</jats:tex-math><mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><mml:mrow><mml:mrow><mml:mi>𝔰</mml:mi><mml:mi>𝔲</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\\\"true\\\" form=\\\"prefix\\\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\\\"true\\\" form=\\\"postfix\\\">)</mml:mo></mml:mrow></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> which leads to the Howe dual pair <jats:inline-formula><jats:alternatives><jats:tex-math>(U(n),\\\\mathfrak{su}(1,2))</jats:tex-math><mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><mml:mrow><mml:mo stretchy=\\\"true\\\" form=\\\"prefix\\\">(</mml:mo><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy=\\\"true\\\" form=\\\"prefix\\\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\\\"true\\\" form=\\\"postfix\\\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi>𝔰</mml:mi><mml:mi>𝔲</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\\\"true\\\" form=\\\"prefix\\\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\\\"true\\\" form=\\\"postfix\\\">)</mml:mo></mml:mrow><mml:mo stretchy=\\\"true\\\" form=\\\"postfix\\\">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>.</jats:p>\",\"PeriodicalId\":355998,\"journal\":{\"name\":\"SciPost Physics Proceedings\",\"volume\":\"32 5\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-11-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SciPost Physics Proceedings\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.21468/scipostphysproc.14.038\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SciPost Physics Proceedings","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.21468/scipostphysproc.14.038","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Unitary Howe dualities in fermionic and bosonic algebras and related Dirac operators
In this paper we use the canonical complex structure \mathbb{J}𝕁 on \mathbb{R}^{2n}ℝ2n to introduce a twist of the symplectic Dirac operator. This can be interpreted as the bosonic analogue of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these Dirac operators is isomorphic to the Lie algebra \mathfrak{su}(1,2)𝔰𝔲(1,2) which leads to the Howe dual pair (U(n),\mathfrak{su}(1,2))(U(n),𝔰𝔲(1,2)).
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