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--- spin7_vanishing [2023/05/12 10:59] – spencer
+++ spin7_vanishing [2023/05/15 10:32] (current) – spencer
@@ Line 19: / Line 19: @@
 <WRAP center round todo 60%>
-**Proof:** Elided, for now
+**Proof:** We have
+\begin{align*}
+D &= c_b \nabla_b \\
+&= -\frac{1}{8} c_a c_a c_b \nabla_b \\
+&= \frac{1}{8} D - \frac{1}{8} \sum_{a \neq b} c_a c_a c_b \nabla_b
+\end{align*}
+so
+$$D = -\frac{1}{7} \sum_{a \neq b} c_a c_a c_b \nabla_b$$
 </WRAP>
 **Lemma 2:** If $\psi \in \Gamma(S_7^+ \otimes E)$, there exist 7 sections $\psi_i \in \Gamma(S_1^+ \otimes E)$ such that $\psi = c(\gamma^i) \psi_i$.
@@ Line 50: / Line 58: @@
 **Lemma 3:** $\ker D = \ker D^7 \cap \ker D^{21}$, the kernels being in $\Gamma(S^+ \otimes E)$.
+We have $|D\psi|^2 = |D^7 \psi|^2 + |D^{21}\psi|^2 + 2\langle D^7\psi, D^{21}\psi\rangle$.
+It suffices (by the assumption that Clifford multiplication commutes with derivatives) to consider $\psi \in S^+_1$; but then the cross term vanishes a priori
 <WRAP center round todo 60%>
-**Proof:** Elided, for now
+do this more carefully later
 </WRAP>