In this question I'll try to avoid using the words *"Borcherd's Lift"* only because I'm not sure in what setting it applies properly. What I will be asking about is sometimes called *"second quantized elliptic genus"* in the physics circles, or perhaps an *"exponential lift of a Jacobi form"*.

One important result is that the elliptic genus $\chi(M_{d} ; q, y)$ of a $d$-dimensional, compact Calabi-Yau manifold $M_{d}$ is a weak Jacobi form of weight zero and index $d/2$. This means the elliptic genus admits a Fourier expansion

$$\chi(M_{d}; q,y) = \sum_{n \geq 0, k \in \mathbb{Z}} c(n, k) q^{n} y^{k},$$

where for a fixed power of $q$, finitely many powers of $y$ contribute. There is also symmetry under $y \to 1/y$. One can use the coefficients of the elliptic genus to produce a infinite product of the form

$$\prod_{m,n,k} \big(1-p^{m}q^{n}y^{k}\big)^{-c(nm, k)}.$$

It is known (https://arxiv.org/abs/math/9906190) that if $M_{d}$ Is Calabi-Yau, then this infinite product will be a Seigel modular form, at least up to some factor. Therefore, **weight zero** (arbitrary index) Jacobi forms seem to play a special role, at least in this sense.

However, **index one** (arbitrary weight) Jacobi forms also appear to have a special role in relation to Siegel modular forms, as indicated in the following figure:

*(Figure taken from Eichler and Zagier, Theory of Jacobi Forms, page 4)*

I'm specifically hoping someone can survey some results for me about the "lifting" of index one Jacobi forms. I have certain weight $2k$, index one Jacobi forms $\varphi_{2k,1}$. Do we know precisely which Siegel modular forms these will lift to? Moreover, assuming we know where each $\varphi_{2k, 1}$ goes to, do we know where a formal linear combination

$$\sum_{k=-1}^{\infty} \lambda^{2k} \varphi_{2k, 1}$$

will lift to? Here, $\lambda$ is a formal variable.

I would also be keenly interested if anyone has insight into why weight zero Jacobi forms, as well as index one Jacobi forms play seemingly such a special role.