One can consider the Calabi-Yau threefold K3$\times E$ where the Donaldson-Thomas theory is conjectured to be the (inverse of) the Igusa cusp form $\chi_{10}(q,y,p)$. The variables $q,y,p$ aren't conventional, I just want to emphasize that it is a three-variable automorphic form. So we have

$$Z^{\text{DT}}_{K3\times E} = \frac{1}{\chi_{10}(q,y,p)}.$$

I've been working with a compact, smooth Calabi-Yau threefold $X$ who has three Kahler classes $d_{1}, d_{2}, d_{3}$. One can compute an equality of the form

$$\frac{1}{2} \log\bigg( \frac{1}{\chi_{10}(q,y,p)}\bigg) = F^{\text{GW}, 1}_{X}(d_{1}, d_{2}, d_{3})$$

where $F^{\text{GW}, 1}_{X}$ is actually the **genus one** Gromov-Witten potential. There is a non-trivial change of variables between the three parameters on each side.

Now, this could be an accidental thing. However, I know string theorists use duality to convert a "hard" problem into an "easy" one. Well the lefthand side of the above equation is a hard computation; it's the full partition function on K3$\times E$. The righthand side though is simply a "one-loop perturbative computation" in Gromov-Witten theory, as a physicist might say. This is (relatively) easy.

So my question is: **is there possibly a string duality lurking here?** If so, are there any more details which jump to anyone's mind? I'm a little hesitant, because I know GW and DT theories rightly belong in *topological string theory* and the web of dualities seems to correspond to the full, physical string theories. But it looks very, very suggestive to me.