Possible string duality example?

Question

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.

Answer 1

This is indeed a "duality" here. In fact it is an example of S-duality (A-model - B -model duality). I will restrict to the case where $E = T^2$

The Igusa cusp form appears in physics in a very interesting context. Consider the compactification of type II string theory on $K3 \times T^2$. The resulting 4d $\mathcal N = 4$ theory has two kinds of extremal BPS representations which are the 1/2-BPS and the 1/4-BPS representations. These states arise when D-branes wrap some combination of the cycles of $K3\times T^2$. The 1/2-BPS states are and they are "counted" by $\dfrac{1}{\eta (\tau)^{24}}$. Then there are the 1/4-BPS objects which are counted by the Borcherds lift of the elliptic genus of the K3 surface which is (surprise surprise) the inverse of the Igusa cusp form. This was first studied in the context of string theory by Dijkgraaf, Verlinde and Verlinde. A more comprehensive review of the Igusa cusp form in string theory are the review by Kawai and the paper by Zagier.et.al.

How does this relate to Donaldson theory? The DT invariants are precisely the counts of these 1/4-BPS states in a different string frame. (There is the string-string duality between type II string theory and the heterotic string theory studied by Aspinwall, and then there is the duality between the type IIA frame and the type IIB frame). In topological string theory terms, we have the A model and the B model. One of the relation between these two models is that DT invariants, which count perverse sheaves therey physically counting the bound states of these D-branes wrapping the cycles of the Calabi-Yau in model B, are precisely the Gromov-Witten invariants in model A. This relation was shown in two papers by Nekrasov, Pandharipandhe et.al.

We come to the final attribute which is the genus 1 GW invariants. Let's start with the fact that all $\mathcal N = 4$ theories can be expressed as $\mathcal N =2$ theories. This allows us to endow a prepotential to these theories. In this special case at hand, the prepotential receives contributions only from genus-1 terms i.e. 1-loop exactness. Hence all the BPS states that are expressed in terms of GW invariants for a $K3 \times T^2$ compactifications necessarily come from genus 1 terms.

So to give you the good news: You were absolutely right to suspect a deep relation here. The bad news: You're about a decade and a half too late.