Thanks a lot. Indeed, it was (4.23) of DMZ which got me thinking about this. But when they write $J_{k,1}$ I believe they mean holomorphic Jacobi forms. Can I clarify that you're claiming the weak Jacobi forms (what DMZ call $\tilde{J}_{k,1}$) are isomorphic to the weakly holomorphic modular forms of weight $k-1/2$ with respect to the congruence subgroup? If so, I assume it is still induced by sending $\varphi$ to $\sum_{\Delta} c(\Delta)q^{\Delta}$, where now $\Delta$ may take negative values.

Thanks for the comment. You seem to state something for general index $m$, so why exactly is $m=1$ special in this example? What you're talking about is perhaps that "middle" isomorphism in the Figure of Eichler and Zagier?

I see; but if we should only be using the isogenies to genus 1 curves, then what canonical bundle were you referring to? If you looks at the general formula for a Hecke operator acting on a weak Jacobi form, of possibly non-zero weight, there is this extra factor that looks like it might be related to the canonical bundle of a genus $g$ curve, as you mention.

Thanks, that's extremely helpful :) So if I'm understanding your last paragraph correctly, if I want to consider the case of non-zero weight $2g-2$, I should study degree $n$ isogenies to a genus $g$ curve, and account for the term coming from the canonical bundle?

Thanks! I am too used to thinking about ideal sheaves in DT theory. The Calabi-Yau threefold here should be the fibrations $A_{n} \to \mathbb{P}^{1}$, correct? In other words, along the lines of the last comment, counting $\text{SU}(n+1)$ instantons on $A_{r-1}$ should be equivalent to studying rank $r$ DT-theory on $A_{n} \to \mathbb{P}^{1}$.

Thanks for your answer! I've been in the process of trying to understand your papers on quiver gauge theories and instantons. I was hoping I could clarify what you meant by "higher rank DT theory?" I know what rank is on the gauge theory side, but not the string theory. It sounds like you mean we take generalized DT invariants for the same CY3 or something like that. We can vary the rank $r$ of the gauge group and the $N$ in $A_{N}$ to get instanton moduli spaces. So all these theories have generating functions equal to (generalized?) DT theory? Unfortunately, I couldn't get your Oxford notes

So if I'm considering a generating function of something like elliptic genera of moduli space of $U(1)$ instantons on $A_{M-1}$, then I really have a gauge theory partition function on $A_{M-1}$ as opposed to $\mathbb{C}^{2}$? Like you say, this is where the ALE space is the spacetime itself. So I take it these crazy product formulas out there should relate Yang-Mills theory on $A_{M-1}$ spaces to topological string theories.