Fermion Representations from the Metric Bundle: Clifford Algebras and the Pati-Salam Spinor
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| Lingua: | inglese |
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2026
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| _version_ | 1866901774901706752 |
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| author | Austermann, Sloan |
| author_facet | Austermann, Sloan |
| contents | <p><strong>Paper 2 of 6</strong> in the Metric Bundle Programme.</p><p>In a companion paper, we showed that the DeWitt supermetric on the space of Lorentzian metrics over four-dimensional spacetime has signature (6,4), yielding the Pati-Salam gauge group \SU(4) × \SU(2)_L × \SU(2)_R as the maximal compact subgroup of \SO(6,4). In this paper, we derive the fermion content of the theory from the Clifford algebra of the positive-norm sector of the normal bundle. The Clifford algebra \Cl(\R^6) \otimes \C ≅ M_8(\C) acts on an eight-dimensional spinor \C^8, which decomposes under the chirality operator as \C^8 = S^+ ⊕ S^- = \mathbf{4} ⊕ \bar{\mathbf{4}} of \SU(4) ≅ \Spin(6). We prove that the \SU(3) subalgebra defined as the centraliser of a complex structure J in \mathfrak{so}(6) coincides exactly with the \SU(3) from the Pati-Salam breaking \SU(4) → \SU(3) × \U(1)_{B-L}, with the \U(1) charge reproducing the baryon-minus-lepton number in the ratio +1/3 : -1 (triplet : singlet). Combined with the (\mathbf{2},\mathbf{1}) ⊕ (\mathbf{1},\mathbf{2}) spinor of \Spin(4) ≅ \SU(2)_L × \SU(2)_R, the full representation is the \mathbf{16} of \Spin(10) ⊃ \Spin(6) × \Spin(4): one complete generation of Standard Model fermions including a right-handed neutrino, with all hypercharges correctly reproduced by Y = (B{-}L)/2 + T_{3R}.</p><p>Part of a six-paper series deriving the Pati-Salam gauge group, fermion content, gauge dynamics, anomaly cancellation, and the three-generation structure from the geometry of the metric bundle Y<sup>14</sup> = Met(X<sup>4</sup>).</p> <p><strong>v2 changes (March 2026):</strong> Added \usepackage{slashed} for correct rendering of Dirac operator notation.</p> <p><strong>v3 changes (March 2026):</strong> v3: Updated Discussion with forward references to Papers IV-VI (anomaly cancellation verified, three-generation conjecture, Yukawa structure). Added Paper4/Paper5/Paper6 bib entries.</p> <p><strong>v4 changes (March 2026):</strong> v4: Added Sp macro (compilation fix). Recompiled.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18946559 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Fermion Representations from the Metric Bundle: Clifford Algebras and the Pati-Salam Spinor Austermann, Sloan metric bundle Clifford algebra Pati-Salam model fermion representations Spin(10) spinor Standard Model fermions right-handed neutrino <p><strong>Paper 2 of 6</strong> in the Metric Bundle Programme.</p><p>In a companion paper, we showed that the DeWitt supermetric on the space of Lorentzian metrics over four-dimensional spacetime has signature (6,4), yielding the Pati-Salam gauge group \SU(4) × \SU(2)_L × \SU(2)_R as the maximal compact subgroup of \SO(6,4). In this paper, we derive the fermion content of the theory from the Clifford algebra of the positive-norm sector of the normal bundle. The Clifford algebra \Cl(\R^6) \otimes \C ≅ M_8(\C) acts on an eight-dimensional spinor \C^8, which decomposes under the chirality operator as \C^8 = S^+ ⊕ S^- = \mathbf{4} ⊕ \bar{\mathbf{4}} of \SU(4) ≅ \Spin(6). We prove that the \SU(3) subalgebra defined as the centraliser of a complex structure J in \mathfrak{so}(6) coincides exactly with the \SU(3) from the Pati-Salam breaking \SU(4) → \SU(3) × \U(1)_{B-L}, with the \U(1) charge reproducing the baryon-minus-lepton number in the ratio +1/3 : -1 (triplet : singlet). Combined with the (\mathbf{2},\mathbf{1}) ⊕ (\mathbf{1},\mathbf{2}) spinor of \Spin(4) ≅ \SU(2)_L × \SU(2)_R, the full representation is the \mathbf{16} of \Spin(10) ⊃ \Spin(6) × \Spin(4): one complete generation of Standard Model fermions including a right-handed neutrino, with all hypercharges correctly reproduced by Y = (B{-}L)/2 + T_{3R}.</p><p>Part of a six-paper series deriving the Pati-Salam gauge group, fermion content, gauge dynamics, anomaly cancellation, and the three-generation structure from the geometry of the metric bundle Y<sup>14</sup> = Met(X<sup>4</sup>).</p> <p><strong>v2 changes (March 2026):</strong> Added \usepackage{slashed} for correct rendering of Dirac operator notation.</p> <p><strong>v3 changes (March 2026):</strong> v3: Updated Discussion with forward references to Papers IV-VI (anomaly cancellation verified, three-generation conjecture, Yukawa structure). Added Paper4/Paper5/Paper6 bib entries.</p> <p><strong>v4 changes (March 2026):</strong> v4: Added Sp macro (compilation fix). Recompiled.</p> |
| title | Fermion Representations from the Metric Bundle: Clifford Algebras and the Pati-Salam Spinor |
| topic | metric bundle Clifford algebra Pati-Salam model fermion representations Spin(10) spinor Standard Model fermions right-handed neutrino |
| url | https://doi.org/10.5281/zenodo.18946559 |