This post is closely related to the post Localization principle in supersymmetry and can be considered as a continuation of it, although independent.

In § 9.3 of the book "Mirror symmetry" (K. Hori et al.) the authors formulate the following general localization principle for computation of integrals over supermanifolds: Assume there is some supersymmetry transformation of variables. Then the integral becomes localized on the field configurations for which the fermionic variables are invariant under the supersymmetry.

I would like to understand this principle in the following example below (which slightly generalizes the example in § 9.3 in the book) and how it should be compatible with Theorem 1 of the paper "Supersymmetry and Localization" by Schwarz and Zaboronsky, see arxiv.org/abs/hep-th/9511112 In example below the vector field $Q$ is chosen such that the integral should be localized over the empty set, i.e. it should vanish.

Let $M^{1|2}$ be the supermanifold with one even coordinate $x$ and two odd ones $\psi_1,\psi_2$. Let $Q$ be an odd vector field on $M$ and $F(x,\psi_1,\psi_2)$ be a function on $M$ defined as follows: \begin{eqnarray*} Q:=\psi_1\partial_x +C(x)\partial_{\psi_2},\\ F(x,\psi_1,\psi_2):=f(x)+\frac{f'(x)}{C(x)}\psi_1\psi_2, \end{eqnarray*} where $f(x),C(x)$ are functions depending on $x$ only, and $C$ never vanishes. It is easy to see that \begin{eqnarray*} Q(F)=0,\\ Q \mbox{ preserves the measure }dx \, d\psi_1\, d\psi_2. \end{eqnarray*} The last condition means that for any function $G(x,\psi_1,\psi_2)$ one has $$\int Q(G)dx \, d\psi_1\, d\psi_2=0.$$ Since $C(x)$ never vanishes by assumption, the integral $\int \exp(-F) dx \, d\psi_1\, d\psi_2$ localizes over the empty set, and should be 0. Explicitly \begin{eqnarray*} \int \exp(-F) dx \, d\psi_1\, d\psi_2=\int \exp(-f(x))(1-\frac{f'(x)}{C(x)}\psi_1\psi_2)dx \, d\psi_1\, d\psi_2=\\-\int \exp(-f(x))\cdot \frac{f'(x)}{C(x)}dx. \end{eqnarray*}

It is clear that the last integral cannot vanish for generic $C$. What is wrong?

Remark 1. In the above book the authors consider a special case of this situation when $$f(x)=\frac{1}{2}H^2(x),\, C(x) =-H(x),$$ where $H$ is non-vanishing. In this case \begin{eqnarray*} \int \exp(-F) dx \, d\psi_1\, d\psi_2=-\int \exp(-f(x))\cdot \frac{f'(x)}{C(x)}dx=\int \exp(-H^2(x))\cdot H'(x)dx. \end{eqnarray*} The last integral vanishes if $M$ is compact.

Remark 2. A sufficient condition for the localization principle to hold formulated in Thm 1 of the above mentioned paper by Schwarz and Zaboronsky is that the vector field $Q^2$ generates a one-parameter group of diffeomorphisms of $M$ contained in a compact group. In our (more general) case $$Q^2= C'(x)\psi_1\partial_{\psi_2}.$$ As far as I understand it does not generate a compact group unless $C'=0$. But this is not the case even in the example from the book.

Thus I do not understand on the conceptual level why the localization principle does work in the example from the book, but does not work in a slightly more general situation. Any feedback is welcome.