TFPT scientific source excerpt
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000155 | \begin{theorem}[Carrier rigidity from the primitive component of the branch Yukawa tensor]
000156 | \label{thm:yukawa-forces-32}
000157 | Let
000158 | \[
000159 | E=E_-\oplus E_+
000160 | \]
000161 | be the two-factor carrier and let
000162 | \[
000163 | G_{\mathrm{car}}:=S\bigl(U(E_-)\times U(E_+)\bigr),
000164 | \qquad
000165 | S^+=\Lambda^{\mathrm{even}}E.
000166 | \]
000167 | Let $\mathbb Y_{\mathrm{br}}$ denote the branch Yukawa tensor induced by the canonical transport
000168 | kernel and rigid family holonomy of
000169 | \TFPTxref{thm:exact-transport-closure,thm:rigid-family-local-system}. Decompose its carrier-homogeneous
000170 | components as trilinear maps of the form
000171 | \[
000172 | B_{p,q;r,t}:
000173 | (\Lambda^p E_-\otimes \Lambda^q E_+)
000174 | \otimes
000175 | (\Lambda^r E_-\otimes \Lambda^t E_+)
000176 | \otimes E_+
000177 | \longrightarrow
000178 | \mathbb C.
000179 | \]
000180 | Let the primitive component of $\mathbb Y_{\mathrm{br}}$ be the unique indecomposable generator of
000181 | \Cref{def:primitive-yukawa-generator,lem:lowest-primitive-yukawa-generator} with two nontrivial
000182 | fermionic legs. Then, up to exchanging the two fermionic legs, that primitive component is
000183 | necessarily of type
000184 | \[
000185 | (E_-\otimes E_+) \otimes \Lambda^2 E_- \otimes E_+
000186 | \longrightarrow
000187 | \Lambda^3 E_-\otimes \Lambda^2 E_+\cong\mathbb C,
000188 | \]
000189 | and therefore
000190 | \[
000191 | \dim E_+=2,
000192 | \qquad
000193 | \dim E_-=3.
000194 | \]
000195 | Consequently
000196 | \[
000197 | \dim \Lambda^2 E_+=1,
000198 | \qquad
000199 | \Lambda^2 E_- \cong E_-^\vee,
000200 | \qquad
000201 | \dim \Lambda^{\mathrm{even}}E=16.
000202 | \]
000203 | Hence the rigid carrier is
000204 | \[
000205 | E=E_3\oplus E_2,
000206 | \qquad
000207 | X=-2P_3+3P_2,
000208 | \qquad
000209 | Y=-\frac13 P_3+\frac12 P_2.
000210 | \]
000211 | \end{theorem}
000212 | 
000213 | \begin{proof}
000214 | By \TFPTxref{cor:bosonic-rank-two}, the compact Higgs index has already fixed
000215 | \[
000216 | \dim E_+=2.
000217 | \]
000218 | Write the primitive nonzero Yukawa component with two nontrivial fermionic legs as
000219 | \[
000220 | B_{p,q;r,t}:
000221 | (\Lambda^p E_-\otimes \Lambda^q E_+)
000222 | \otimes
000223 | (\Lambda^r E_-\otimes \Lambda^t E_+)
000224 | \otimes E_+
000225 | \longrightarrow
000226 | \mathbb C
000227 | \]
000228 | and note that \Cref{lem:lowest-primitive-yukawa-generator} forces primitiveness in the invariant-ring
000229 | sense to have
000230 | \[
000231 | (q,t)=(1,0),
000232 | \qquad
000233 | (p,r)=(1,2)
000234 | \]
000235 | up to exchanging the two fermionic legs. Thus the primitive carrier component is exactly
000236 | \[
000237 | (E_-\otimes E_+) \otimes \Lambda^2 E_- \otimes E_+.
000238 | \]
000239 | Its negative factor contracts canonically as
000240 | \[
000241 | E_- \otimes \Lambda^2 E_- \longrightarrow \Lambda^3 E_-,
000242 | \]
000243 | so scalarity forces $\Lambda^3 E_-$ to be the top exterior power. Hence
000244 | \[
000245 | \dim E_-=3.