Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917738372399104 |
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| author | Liu, Chengcheng Majid, Shahn |
| author_facet | Liu, Chengcheng Majid, Shahn |
| contents | Using the framework of quantum Riemannian geometry, we show that gravity on the product of spacetime and a fuzzy sphere is equivalent under minimal assumptions to gravity on spacetime, an $su_2$-valued Yang-Mills field $A_{μi}$ and real-symmetric-matrix valued Liouville-sigma model field $h_{ij}$ for gravity on the fuzzy sphere. Moreover, a massless real scalar field on the product appears as a tower of scalar fields on spacetime, with one for each internal integer spin $l$ representation of $SU(2)$, minimally coupled to $A_{μi}$ and with mass depending on $l$ and the fuzzy sphere size. For discrete values of the deformation parameter, the fuzzy spheres can be reduced to matrix algebras $M_{2j+1}(C)$ for $j$ any non-negative half-integer, and in this case only integer spins $0\le l\le 2j$ appear in the multiplet. Thus, for $j=1$ a massless field on the product appears as a massless $SU(2)$ internal spin 0 field, a massive internal spin 1 field and a massive internal spin 2 field, in mass ratio $0,1,\sqrt{3}$ respectively, which we conjecture could arise in connection with an approximate $SU(2)$ flavour symmetry. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_14128 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model Liu, Chengcheng Majid, Shahn High Energy Physics - Theory General Relativity and Quantum Cosmology Quantum Algebra Using the framework of quantum Riemannian geometry, we show that gravity on the product of spacetime and a fuzzy sphere is equivalent under minimal assumptions to gravity on spacetime, an $su_2$-valued Yang-Mills field $A_{μi}$ and real-symmetric-matrix valued Liouville-sigma model field $h_{ij}$ for gravity on the fuzzy sphere. Moreover, a massless real scalar field on the product appears as a tower of scalar fields on spacetime, with one for each internal integer spin $l$ representation of $SU(2)$, minimally coupled to $A_{μi}$ and with mass depending on $l$ and the fuzzy sphere size. For discrete values of the deformation parameter, the fuzzy spheres can be reduced to matrix algebras $M_{2j+1}(C)$ for $j$ any non-negative half-integer, and in this case only integer spins $0\le l\le 2j$ appear in the multiplet. Thus, for $j=1$ a massless field on the product appears as a massless $SU(2)$ internal spin 0 field, a massive internal spin 1 field and a massive internal spin 2 field, in mass ratio $0,1,\sqrt{3}$ respectively, which we conjecture could arise in connection with an approximate $SU(2)$ flavour symmetry. |
| title | Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Quantum Algebra |
| url | https://arxiv.org/abs/2312.14128 |