Moduli space of genus $0$ degree $d$ maps in a quintic Calabi-Yau threefold $X$, written as $\overline{\mathcal{M}}_{0,0}(X,[d])$, can be embedded in the corresponding moduli space of $\mathbb{P}^4$, i.e. $\overline{\mathcal{M}}_{0,0}(\mathbb{P}^4,[d])$. The latter is an orbifold. There is orbibundle $E_{0,d}$ on $\overline{\mathcal{M}}_{0,0}(\mathbb{P}^4,[d])$ whose fiber over every map $[u\colon \Sigma\to X]$ is $H^0(u^*TX)$. If $F\in \mathcal{O}_{\mathbb{P}^4}(5)$ is the defining equation of $X$, it gives us a section $$ s_F\colon \overline{\mathcal{M}}_{0,0}(\mathbb{P}^4,[d])\to E_{0,d},\quad s_F(u)=u^*F $$ whose zero set is $\overline{\mathcal{M}}_{0,0}(X,[d])$. Thus, via theory of Kuranishi structures (or other similar approaches), we can think of the tuple $(E_{0,d},\overline{\mathcal{M}}_{0,0}(\mathbb{P}^4,[d],s_F)$ as an abstract kuranishi structure on $\overline{\mathcal{M}}_{0,0}(X,[d])$. Let $N_{0,d}$ be the euler class of this bundle (which is just a number).
On the otherhand, we can directly build a set of natural Kuranishi structures on $\overline{\mathcal{M}}_{0,0}(X,[d])$ which allows us to associate a VFC to $\overline{\mathcal{M}}_{0,0}(X,[d])$. This construction works for every moduli space of maps. In this case, VFC is again a number $N'_{0,d}$ and these are genus $0$ GW invariants of quintic.
The well-known calculation of GW invariants of quintic via localization uses $N_{0,d}=N_{0,d}'$.
**** Question: Where is it shown/outlined that the two are the same? (I am primarily interested in an analytic approach).
**** Some references: In the algebraic case, this is stated as Theorem 26.1.1 in the big Mirror Symmetry book without proof. Cox-Kats (Example 7.1.5.1) gives a scheme of proving that. In the analytic side, Section 5.3 of
http://www.ams.org/journals/jams/2008-21-04/S0894-0347-08-00597-3/S0894-0347-08-00597-3.pdf
outlines a way of proving this in the theory of Kuranishi structures. I am looking for other places that this has been discussed with more details.
