#grandunification — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #grandunification, aggregated by home.social.
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🧮 New preprint — Part II of a series: *Three Gates to a Quark Generation*.
The question. In 6D gauge–Higgs unification, which representations of \(SU(4)\) hold a Standard-Model quark generation as chiral zero modes on \(T^2/\mathbb{Z}_2\)?
The answer, exactly. An irreducible rep with Dynkin labels \((a,b,c)\) works iff
\[(a+2b+3c)\ \text{odd}\ \ \wedge\ \ b\ge 1\ \ \wedge\ \ a+b+c\ge 3.\]
Three gates: arithmetic (the \(\mathbb{Z}_4\) centre charge), geometric (the middle Dynkin node excited), and size. Only 8 reps qualify below dim 400.For the curious. Strip a generation to its skeleton, the “\(\pm\tfrac12\) cell”: \(Q(\mathbf 2,\tfrac16),\, u(\mathbf 1,\tfrac23),\, d(\mathbf 1,-\tfrac13)\), where \(\tfrac16=\tfrac12\bigl(\tfrac23+(-\tfrac13)\bigr)\) puts the doublet at the hypercharge midpoint of its singlets — Pati–Salam's \(Y=T^3_R+\tfrac{B-L}{2}\). Now delete the middle node of \(\circ\!-\!\circ\!-\!\circ\): out comes a Levi \(SU(2)_L\times SU(2)_R\times U(1)\). Its \(SU(2)_L\)-singlet sector is \((b{+}1)\) copies of the Clebsch–Gordan tower \([\tfrac a2]\otimes[\tfrac c2]\); the orbifold's chirality keeps half of each, giving a closed zero-mode count
\[N(a,b,c)=(b{+}1)\,\tfrac{a+c+1}{2}.\]The twist: why \(SU(4)\)? The cell is three charge constraints on a hypercharge with \(\operatorname{rank}(G){-}1\) free parameters — over-determined exactly at rank 3. So \(SU(4)\) is the unique critical rank where the cell is a rigid gate; for \(SU(5),SU(6)\) it dissolves and generic reps admit. Machine-checked exactly to dimension 900.
📄 https://doi.org/10.5281/zenodo.21432628
💻 https://github.com/karlesmarin/su4-sm-cell-criterion#RepresentationTheory #Physics #GaugeTheory #GrandUnification
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🧮 New preprint — Part II of a series: *Three Gates to a Quark Generation*.
The question. In 6D gauge–Higgs unification, which representations of \(SU(4)\) hold a Standard-Model quark generation as chiral zero modes on \(T^2/\mathbb{Z}_2\)?
The answer, exactly. An irreducible rep with Dynkin labels \((a,b,c)\) works iff
\[(a+2b+3c)\ \text{odd}\ \ \wedge\ \ b\ge 1\ \ \wedge\ \ a+b+c\ge 3.\]
Three gates: arithmetic (the \(\mathbb{Z}_4\) centre charge), geometric (the middle Dynkin node excited), and size. Only 8 reps qualify below dim 400.For the curious. Strip a generation to its skeleton, the “\(\pm\tfrac12\) cell”: \(Q(\mathbf 2,\tfrac16),\, u(\mathbf 1,\tfrac23),\, d(\mathbf 1,-\tfrac13)\), where \(\tfrac16=\tfrac12\bigl(\tfrac23+(-\tfrac13)\bigr)\) puts the doublet at the hypercharge midpoint of its singlets — Pati–Salam's \(Y=T^3_R+\tfrac{B-L}{2}\). Now delete the middle node of \(\circ\!-\!\circ\!-\!\circ\): out comes a Levi \(SU(2)_L\times SU(2)_R\times U(1)\). Its \(SU(2)_L\)-singlet sector is \((b{+}1)\) copies of the Clebsch–Gordan tower \([\tfrac a2]\otimes[\tfrac c2]\); the orbifold's chirality keeps half of each, giving a closed zero-mode count
\[N(a,b,c)=(b{+}1)\,\tfrac{a+c+1}{2}.\]The twist: why \(SU(4)\)? The cell is three charge constraints on a hypercharge with \(\operatorname{rank}(G){-}1\) free parameters — over-determined exactly at rank 3. So \(SU(4)\) is the unique critical rank where the cell is a rigid gate; for \(SU(5),SU(6)\) it dissolves and generic reps admit. Machine-checked exactly to dimension 900.
📄 https://doi.org/10.5281/zenodo.21432628
💻 https://github.com/karlesmarin/su4-sm-cell-criterion#RepresentationTheory #Physics #GaugeTheory #GrandUnification
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🧮 New preprint — Part II of a series: *Three Gates to a Quark Generation*.
The question. In 6D gauge–Higgs unification, which representations of \(SU(4)\) hold a Standard-Model quark generation as chiral zero modes on \(T^2/\mathbb{Z}_2\)?
The answer, exactly. An irreducible rep with Dynkin labels \((a,b,c)\) works iff
\[(a+2b+3c)\ \text{odd}\ \ \wedge\ \ b\ge 1\ \ \wedge\ \ a+b+c\ge 3.\]
Three gates: arithmetic (the \(\mathbb{Z}_4\) centre charge), geometric (the middle Dynkin node excited), and size. Only 8 reps qualify below dim 400.For the curious. Strip a generation to its skeleton, the “\(\pm\tfrac12\) cell”: \(Q(\mathbf 2,\tfrac16),\, u(\mathbf 1,\tfrac23),\, d(\mathbf 1,-\tfrac13)\), where \(\tfrac16=\tfrac12\bigl(\tfrac23+(-\tfrac13)\bigr)\) puts the doublet at the hypercharge midpoint of its singlets — Pati–Salam's \(Y=T^3_R+\tfrac{B-L}{2}\). Now delete the middle node of \(\circ\!-\!\circ\!-\!\circ\): out comes a Levi \(SU(2)_L\times SU(2)_R\times U(1)\). Its \(SU(2)_L\)-singlet sector is \((b{+}1)\) copies of the Clebsch–Gordan tower \([\tfrac a2]\otimes[\tfrac c2]\); the orbifold's chirality keeps half of each, giving a closed zero-mode count
\[N(a,b,c)=(b{+}1)\,\tfrac{a+c+1}{2}.\]The twist: why \(SU(4)\)? The cell is three charge constraints on a hypercharge with \(\operatorname{rank}(G){-}1\) free parameters — over-determined exactly at rank 3. So \(SU(4)\) is the unique critical rank where the cell is a rigid gate; for \(SU(5),SU(6)\) it dissolves and generic reps admit. Machine-checked exactly to dimension 900.
📄 https://doi.org/10.5281/zenodo.21432628
💻 https://github.com/karlesmarin/su4-sm-cell-criterion#RepresentationTheory #Physics #GaugeTheory #GrandUnification
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🧮 New preprint — Part II of a series: *Three Gates to a Quark Generation*.
The question. In 6D gauge–Higgs unification, which representations of \(SU(4)\) hold a Standard-Model quark generation as chiral zero modes on \(T^2/\mathbb{Z}_2\)?
The answer, exactly. An irreducible rep with Dynkin labels \((a,b,c)\) works iff
\[(a+2b+3c)\ \text{odd}\ \ \wedge\ \ b\ge 1\ \ \wedge\ \ a+b+c\ge 3.\]
Three gates: arithmetic (the \(\mathbb{Z}_4\) centre charge), geometric (the middle Dynkin node excited), and size. Only 8 reps qualify below dim 400.For the curious. Strip a generation to its skeleton, the “\(\pm\tfrac12\) cell”: \(Q(\mathbf 2,\tfrac16),\, u(\mathbf 1,\tfrac23),\, d(\mathbf 1,-\tfrac13)\), where \(\tfrac16=\tfrac12\bigl(\tfrac23+(-\tfrac13)\bigr)\) puts the doublet at the hypercharge midpoint of its singlets — Pati–Salam's \(Y=T^3_R+\tfrac{B-L}{2}\). Now delete the middle node of \(\circ\!-\!\circ\!-\!\circ\): out comes a Levi \(SU(2)_L\times SU(2)_R\times U(1)\). Its \(SU(2)_L\)-singlet sector is \((b{+}1)\) copies of the Clebsch–Gordan tower \([\tfrac a2]\otimes[\tfrac c2]\); the orbifold's chirality keeps half of each, giving a closed zero-mode count
\[N(a,b,c)=(b{+}1)\,\tfrac{a+c+1}{2}.\]The twist: why \(SU(4)\)? The cell is three charge constraints on a hypercharge with \(\operatorname{rank}(G){-}1\) free parameters — over-determined exactly at rank 3. So \(SU(4)\) is the unique critical rank where the cell is a rigid gate; for \(SU(5),SU(6)\) it dissolves and generic reps admit. Machine-checked exactly to dimension 900.
📄 https://doi.org/10.5281/zenodo.21432628
💻 https://github.com/karlesmarin/su4-sm-cell-criterion#RepresentationTheory #Physics #GaugeTheory #GrandUnification
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🧮 New preprint — Part II of a series: *Three Gates to a Quark Generation*.
The question. In 6D gauge–Higgs unification, which representations of \(SU(4)\) hold a Standard-Model quark generation as chiral zero modes on \(T^2/\mathbb{Z}_2\)?
The answer, exactly. An irreducible rep with Dynkin labels \((a,b,c)\) works iff
\[(a+2b+3c)\ \text{odd}\ \ \wedge\ \ b\ge 1\ \ \wedge\ \ a+b+c\ge 3.\]
Three gates: arithmetic (the \(\mathbb{Z}_4\) centre charge), geometric (the middle Dynkin node excited), and size. Only 8 reps qualify below dim 400.For the curious. Strip a generation to its skeleton, the “\(\pm\tfrac12\) cell”: \(Q(\mathbf 2,\tfrac16),\, u(\mathbf 1,\tfrac23),\, d(\mathbf 1,-\tfrac13)\), where \(\tfrac16=\tfrac12\bigl(\tfrac23+(-\tfrac13)\bigr)\) puts the doublet at the hypercharge midpoint of its singlets — Pati–Salam's \(Y=T^3_R+\tfrac{B-L}{2}\). Now delete the middle node of \(\circ\!-\!\circ\!-\!\circ\): out comes a Levi \(SU(2)_L\times SU(2)_R\times U(1)\). Its \(SU(2)_L\)-singlet sector is \((b{+}1)\) copies of the Clebsch–Gordan tower \([\tfrac a2]\otimes[\tfrac c2]\); the orbifold's chirality keeps half of each, giving a closed zero-mode count
\[N(a,b,c)=(b{+}1)\,\tfrac{a+c+1}{2}.\]The twist: why \(SU(4)\)? The cell is three charge constraints on a hypercharge with \(\operatorname{rank}(G){-}1\) free parameters — over-determined exactly at rank 3. So \(SU(4)\) is the unique critical rank where the cell is a rigid gate; for \(SU(5),SU(6)\) it dissolves and generic reps admit. Machine-checked exactly to dimension 900.
📄 https://doi.org/10.5281/zenodo.21432628
💻 https://github.com/karlesmarin/su4-sm-cell-criterion#RepresentationTheory #Physics #GaugeTheory #GrandUnification
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#AlanGuth - How Do #Particles Explain the #Cosmos?
https://www.youtube.com/watch?v=xLet07q6bvo&ab_channel=CloserToTruth
#Science #Physics #Cosmology #Energy #GUT #GrandUnificationTheories #GrandUnification #LinearAccelerator #LinearAccelerators #ParticleAccelerator #ParticleAccelerators #Inflation #Inflaton #InflatonField #RepulsiveGravity #QuantumPhysics #QuantumMechanics #QM #BigBang #HotBigBang #CloserToTruth #RobertKuhn
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#AlanGuth - How Do #Particles Explain the #Cosmos?
https://www.youtube.com/watch?v=xLet07q6bvo&ab_channel=CloserToTruth
#Science #Physics #Cosmology #Energy #GUT #GrandUnificationTheories #GrandUnification #LinearAccelerator #LinearAccelerators #ParticleAccelerator #ParticleAccelerators #Inflation #Inflaton #InflatonField #RepulsiveGravity #QuantumPhysics #QuantumMechanics #QM #BigBang #HotBigBang #CloserToTruth #RobertKuhn
