# Possible string duality example?

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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.

Some remarks: if you forget two of the three variables then $\chi_{10}$ essentially reduces to the modular form $\Delta$ (capturing genus zero GW invariants of K3) and it is well-known that $log (\Delta)$ is essentially the genus 1 Gromov-Witten potential of an elliptic curve. This looks like a simplified version of the relation you mention. Even in this simpler case, I don't know an easy string duality explanation (there are string theoretic derivations of both sides and they both give $\Delta$ because modular forms often appear but I don't know how to see directly that the reasons why they appear are the same). The fact that GW is part of topological strings is not an issue a priori: they are standard ways to embed topological string theory in the full physical string theory and then to exploit the web of dualities (and this has been done for many purposes). Do you mind explaining what is the CY 3-fold X and what is the non-trivial change of variables? Maybe this could give more insights into the problem.
Thanks a lot for pointing out the connection between $\log(\Delta)$ and the genus one GW invariants on the elliptic curve.  I was unaware of that myself.  I should note, that my relation seems much stronger though, because as you say, $\Delta$ is merely the genus 0 invariants on a K3, while $\log(1/ \chi_{10})$ is like the full topological string partition function on $K3 \times E$.  So I'm guessing, this variable specialization you're talking about probably specializes to genus zero and degree 0 on the elliptic curve factor.
Anyways, roughly speaking $X$ is a Calabi-Yau threefold which is essentially a fibration of $E \times E$ over $\mathbb{P}^{1}$.  But of course, certain fibers degenerate at places.  So I was thinking the relation might come from some duality between $K3 \times E$ and $T^{6}$ or something.  I haven't actually worked out the change of variables properly but you're right, that may shed light on any possible dualities.
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