TFPT scientific source excerpt
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000166 | \begin{proposition}[Hypercharge as an affine form of the seam involution]
000167 | For the minimal carrier,
000168 | \[
000169 | Y=-\frac13 P_3+\frac12 P_2
000170 | =\mathrm{diag}\!\left(-\frac13,-\frac13,-\frac13,\frac12,\frac12\right),
000171 | \]
000172 | and equivalently
000173 | \[
000174 | Y=\frac{1}{12}\mathbf 1+\frac{5}{12}\tau.
000175 | \]
000176 | \end{proposition}
000177 | 
000178 | \begin{proof}
000179 | Using $P_3=(1-\tau)/2$ and $P_2=(1+\tau)/2$ gives
000180 | \[
000181 | -\frac13 P_3+\frac12 P_2
000182 | = -\frac16(1-\tau)+\frac14(1+\tau)
000183 | = \frac{1}{12}\mathbf 1+\frac{5}{12}\tau.
000184 | \]
000185 | \end{proof}
000186 | 
000187 | \begin{theorem}[Global Standard Model theorem]
000188 | The determinant-preserving connected stabilizer of the minimal carrier split is
000189 | \[
000190 | \mathrm{Aut}_0(E_3\oplus E_2,\det=1)
000191 | = S(U(3)\times U(2))
000192 | \cong \frac{SU(3)\times SU(2)\times U(1)_Y}{\mathbb Z_6}.
000193 | \]
000194 | \end{theorem}
000195 | 
000196 | \begin{proof}
000197 | Automorphisms preserving the split and the determinant condition act independently on $E_3$ and $E_2$ with unit determinant on the combined carrier. This gives $S(U(3)\times U(2))$. The covering map from $SU(3)\times SU(2)\times U(1)_Y$ to $S(U(3)\times U(2))$ has the canonical $\mathbb Z_6$ subgroup as kernel, yielding the stated global Standard Model form.
000198 | \end{proof}
000199 | 
000200 | Set
000201 | \[
000202 | S^+ := \Lambda^{\mathrm{even}}(E_3\oplus E_2).
000203 | \]
000204 | 
000205 | \begin{proposition}[Carrier invariants]
000206 | For the minimal carrier,
000207 | \[
000208 | \gamma := \tr_EY^2 = \frac56,
000209 | \qquad
000210 | B := \frac{\rank E_3}{\rank E_2} = \frac32,
000211 | \qquad
000212 | \dim S^+ = 16.
000213 | \]
000214 | \end{proposition}
000215 | 
000216 | \begin{proof}
000217 | The first identity is
000218 | \[
000219 | \tr_EY^2 = 3\left(\frac19\right)+2\left(\frac14\right)=\frac13+\frac12=\frac56.
000220 | \]
000221 | The second is immediate from the split ranks. The third follows from
000222 | \[
000223 | \dim S^+ = \dim \Lambda^{\mathrm{even}}E = 2^{5-1}=16.
000224 | \]
000225 | \end{proof}
000226 | 
000227 | \begin{definition}[Minimal carrier packet]
000228 | The rigid carrier packet of v2.9 is
000229 | \[
000230 | \Cmin := (E_3\oplus E_2,\;Y,\;S^+,\;\gamma,\;B).
000231 | \]
000232 | \end{definition}
000233 | 
000234 | \section{Carrier Matter, Trace Identities, and the Abelian Index}
000235 | 
000236 | \subsection{One chiral family from the positive half-spinor}
000237 | 
000238 | \begin{theorem}[Carrier matter theorem]
000239 | For the minimal carrier $E=E_3\oplus E_2$, the positive half-spinor
000240 | \[
000241 | S^+ := \Lambda^{\mathrm{even}}(E_3\oplus E_2)
000242 | \]
000243 | decomposes as
000244 | \[
000245 | (1,1)_0
000246 | \oplus
000247 | (3,2)_{1/6}
000248 | \oplus
000249 | (\bar 3,1)_{-2/3}
000250 | \oplus
000251 | (\bar 3,1)_{1/3}
000252 | \oplus
000253 | (1,2)_{-1/2}
000254 | \oplus
000255 | (1,1)_1.
000256 | \]
000257 | Thus $S^+$ carries exactly one Standard Model family with a right-handed neutrino:
000258 | \[
000259 | \nu^c,\ Q,\ u^c,\ d^c,\ L,\ e^c.
000260 | \]
000261 | \end{theorem}
000262 | 
000263 | \begin{proof}
000264 | The even exterior powers split as
000265 | \[
000266 | \Lambda^{\mathrm{even}}E
000267 | =
000268 | \Lambda^0 E \oplus \Lambda^2 E \oplus \Lambda^4 E.
000269 | \]
000270 | Now
000271 | \[
000272 | \Lambda^0 E = (1,1)_0,
000273 | \]
000274 | \[
000275 | \Lambda^2 E = \Lambda^2 E_3 \oplus (E_3\otimes E_2)\oplus \Lambda^2 E_2
000276 | = (\bar 3,1)_{-2/3}\oplus (3,2)_{1/6}\oplus (1,1)_1,
000277 | \]
000278 | and $\Lambda^4E \cong E^\ast$ because $\det E$ is trivial in the $S(U(3)\times U(2))$ form, so
000279 | \[
000280 | \Lambda^4E \cong (\bar 3,1)_{1/3}\oplus (1,2)_{-1/2}.
000281 | \]
000282 | Collecting the terms gives the stated multiplets.