Lemma: Odd tangency

For $u\in {\mathcal{C}}_W$, the free variation is an odd tangent vector to ${\mathcal{C}}_W$. Consequently $s_0{\mathcal{N}}_W\subseteq {\mathcal{N}}_W$ for the classical vanishing space defined in Section sec:ideal.

Proof. On the shell,

Also $\square b=D\square \gamma =0$, so $s_0(\square \bar{c})=i\square b=0$. The other free equations are unchanged. The additional restriction satisfies $s_0(D^{\ast }b)=0$. These statements are interpreted with an external odd infinitesimal parameter. Differentiating a polynomial identity on this linear restriction space proves preservation of its vanishing space.