Let $M = \{ G(x) = 0 \} \subseteq \mathbb{P}^4$ be a quintic Calabi-Yau and $\mathbf{e} \in H^2(M, \mathbb{Z})$ such that $\int_M \mathbf{e}^3 = 5$. Then as $t \gg 1$:

$$ \int_M e^{n \mathbf{e}} e^{-\frac{t\mathbf{e}}{2\pi i}} \left( 1 + \frac{5}{6} \mathbf{e}^2 \right)^{1/2} = - \frac{5}{12} \left( \frac{t}{2\pi i} - n\right) \left( 2\left( \frac{t}{2\pi i} - n\right)^2 - 5 \right) \tag{$\ast$}$$

How does a high-frequency integral like this get computed? This could be steepest descent or stationary phase. In that case what are the Leftschetz thimble and Morse flow? Perhaps set $G$ bo the the Fermat quintic or something for an example.

**Question** Because there are so many new elements for me here: the surface in $\mathbb{P}^4$, digesting what it means $\int_M \mathbf{e}^3 =5$. In a deep way, this is hardly more than an Laplace transform. I would like help seeing how steepest descent / stationary phase is implemented and computed here.

The original paper talks about the pullback of the line bundle $\mathcal{O}_{\mathbb{P}^4}(n)$ to $\mathcal{O}_M(n)$ and details of $\hat{A}$-genus that I am relegating to a separate question.

Withholding the physics context, the left side looks like particular case of a more general result, but this is already an involved computation. I am struggling to see even this much and connections to more classical mathematics.

**Tongue-in-cheek remark** if the moments on the left side could be computed, then we get some analogue of Stirling formula $\log n! \approx n \log n$.

**Clarifications**

The notation $(1 + \frac{5}{6} \mathbf{e}^2)^{1/2}$ is taken from an arXiv.hep-th paper from 2013 which in turn cites others. One possible interpretation is to use the Taylor series:

$$(1 + \frac{5}{6} \mathbf{e}^2)^{1/2} = 1 + \binom{\tfrac{1}{2}}{1} \left(\tfrac{5}{6}\right)^1 \mathbf{e}^2 + \binom{\tfrac{1}{2}}{1} \left(\tfrac{5}{6}\right)^2 \mathbf{e}^4 + \binom{\tfrac{1}{2}}{1} \left(\tfrac{5}{6}\right)^3 \mathbf{e}^6 + \dots $$

The remaining terms are $0$ since $M$ is 6-dimensional. At least that's the gist of one of the comments.

Another comment suggests the integral $\ast$ is

**exact**: an equality for all values of $t,n$.Also $\mathbb{e}$ is just a 2-form like $x \, dx \vee dy$ restricted to the manifold $M$. How are we assured these exists one with $\int_M \mathbb{e}^3 = 5$?