I am working with the Berline–Vergne localization formula (from Heat Kernels and Dirac Operators): For a compact Lie group $G$ acting on a compact manifold $M$ and an equivariantly closed form $\alpha$, their theorem (7.13) states that for $X \in \mathfrak{g}$ and $Y\in \mathfrak{g}_{0}=\{Y\in\mathfrak{g}:[X,Y]=0\}$ and sufficiently close to $X$, the integral of $\alpha(Y)$ localizes to the zero set $M_0$ of the fundamental vector field $X_M$: $$\int_{M}\alpha(Y)=\int_{M_{0}}(-2\pi)^{\operatorname{rk}(\mathcal{N}/2)}\frac{\alpha(Y)}{\chi_{\mathfrak{g}_{0}}(\mathcal{N})(Y)}$$ where $\chi_{\mathfrak{g}_{0}}(\mathcal{N})$ is the equivariant Euler form of the normal bundle $\mathcal{N}$ of $M_{0}$.
In other references (e.g. theorem 30.2 of Introductory Lectures on Equivariant Cohomology), I frequently encounter a more abstract localization formula (when $G=T$ is a torus) where the integral is computed directly over the fixed point set $F$ of the entire group action, often written as: $$\int_{M}\phi=\int_{F}\frac{\iota_{F}^{*}\phi}{e^{T}(N)}$$
My principal question is: How does one derive the second, "global" fixed-point formula from the more general Berline-Vergne formula, which explicitly depends on a choice of $\boldsymbol{X \in \mathfrak{g}}$.
I know that the torus plays an important role because it's abelian and hence its Lie algebra too, but this doesn't simplify things directly for me.