# Possible string duality example?

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.

• What do you mean by "three Kähler classes"? Do you mean that your threefold is hyper-Kähler?
– Danu
Commented Aug 19, 2017 at 13:47
• @Danu No, what I meant is that I'm only "counting curves" in three distinct Kahler classes in $X$. In some geometry with $h^{1,1}$ at least three, you can introduce formal variables $d_{1}, d_{2}, d_{3}$ which track the degree along each of the three curves of interest. Of course, there could be more curve classes, but you might choose to have "degree zero" on them, which amounts to essentially setting their formal variables to zero. Commented Aug 19, 2017 at 14:31
• Thanks for the answer. I must admit that I'm not really acquainted with questions of enumerative geometry and the physical approaches to them, but if you'd like to explain some of it to me you can find me in Physics Chat.
– Danu
Commented Aug 19, 2017 at 14:40

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.

• Yes I do recall that. However, I actually do not have a comment about it right now because I'm still looking into it. I only have a comment to make when $E = T^2$ where the genus 1 correction is not the log of the Igusa but rather the log of the Dedekind eta. Do you happen to have a paper where the log of the Igusa appears in the $g = 1$ prepotential? Commented May 19, 2020 at 4:31
• The reason I restricted to $T^2$ is that it has a very well known story and a case where $K3 \times E$ (generic $E$) shows up is usually a CHL or a quotient of this $K3 \times T^2$ story in which case a lot of the duality sort of follows through. Commented May 19, 2020 at 4:34
• Yes, in my MSc thesis from last year (arxiv.org/pdf/1905.07085.pdf) see equation (7.55). The background Calabi-Yau geometry is a space Jim Bryan named the "banana manifold." It's generically fibered over $\mathbb{P}^{1}$ with products of elliptic curves $E \times E$. And from my thesis, the genus 1 GW potential in the three fiber classes is a log of the Igusa. Would be cool if there was some physics duality explaining this relationship! Commented May 19, 2020 at 8:15
• Oh! Perhaps we should discuss this beyond the scope of this site? This is actually closely related to a current research project of mine and I would be happy to discuss this more with you. Coincidentally, I think we might have already crossed paths before. Commented May 19, 2020 at 23:11
• That would be great! I'd be happy to chat about this, or anything related. Want to email me at spietro (at) math.ubc.ca? Commented May 20, 2020 at 4:07