# Supersymmetric Localization (Mirror Symmetry)

I'm reading Chapter 9 of Mirror Symmetry book. As you can see in eq. (9.30) his model for SUSY is

\begin{align} \delta_\epsilon X &=\epsilon^1\psi_1 + \epsilon^2\psi_2\\ \delta\psi_1 &= \epsilon^2\partial h\\ \delta\psi_2 &=-\epsilon^1\partial h. \end{align}\tag{9.30}

I understand that if $\partial h\neq 0$ it is possible to see that partition function vanishes, in particular, there is a total divergence:

$$\int \mathrm{d}\hat{X}e^{\frac{1}{2}(\partial h(\hat{X}))^2}\frac{\partial^2 h(\hat{X})}{(\partial h(\hat{X}))^2}. \tag{*}$$

The implication that a total divergence yields vanishing integral, it is a bit strange, because, since every function that admit a primitive is a total derivative. What is the meaning of that implication? Then why $(*)$ vanishes? It seems to me that, in general, $$\int \mathrm{d}(h')\frac{e^{\frac{1}{2}(h')^2}}{(h')^2}\neq 0.$$

Then, again, when he claims that if there is a $x_0$ such that $h'(x_0)=0$ we can use again the argument of the total divergence, why doesn't he keep in account that, removing a small neighborhood of $x_0$, give us some contribution at the border of that neighborhood (two points)?

1. OP essentially wrote (v2):

The implication that a total divergence $f=F^{\prime}$ yields vanishing integral $\int_{\mathbb{R}} \! dx ~f(x)=0$ is a bit strange, because, since every function $f$ that admit a primitive $F$ is a total derivative $f=F^{\prime}$.

Answer: Well, the devil is in the detail. Ref. 1 starts with a function $F$ that is assumed$^1$ to vanishes at $x=\pm \infty$: $$F(x\!=\!\pm \infty)~=~0. \tag{A}$$ If we, on the other hand, instead start from a function $f$, then all its primitives $F$ will generically not satisfy condition (A).

2. OP essentially wrote (v2):

Why doesn't Ref. 1 keep in account that, removing a small neighborhood of $x_0$, give us some contribution at the border of that neighborhood (two points)?

Answer: The neighborhood may be chosen as small as we want, thereby localizing the path integral $Z$ at the point $x_0$.

$^1$ Ref. 1 does not seem to explicitly mentioning assumption (A), but that is what is meant. Technically, it is sufficient if $$F(x\!=\!- \infty)~=~F(x\!=\! \infty). \tag{B}$$
• I do not understand two points: 1) Where is (hidden) the vanishing assumption of $F$? 2) Whatever you shrink the neighborhood near $x_0$ you are remaining with two boundary contributions. How can one cure that?