Robbert Dijkgraaf said,1
concerning the simplest
Calabi–Yau space, the [quintic](https://en.wikipedia.org/wiki/Quintic_threefold):



> 
> "A classical result from the 19th century states that the number of lines — degree-one curves — is equal to 2,875. The number of degree-two curves was only computed around 1980 and turns out to be much larger: 609,250. But the number of curves of degree three required the help of string theorists."
> 
> 
> 


I gather from [OEIS sequence A076912](https://oeis.org/A076912) that
the number is 317,206,375.



> 
> ***Q***. Can anyone point me to (or describe) how the degree-3 count
>  was settled with "the help of string theorists"?
> 
> 
> 




---


1Robbert Dijkgraaf. "Quantum Questions Inspire New Math."
[*Quanta*](https://www.quantamagazine.org/how-quantum-theory-is-inspiring-new-math-20170330/).
*The Best Writing in Mathematics 2018*, Ed. M. Pitici. p.80. Princeton.
[Publisher's link](https://press.princeton.edu/titles/14178.html).



Degree-3 curves on the Calabi–Yau quintic
The physicists [Candelas, De La Ossa, Green, Parkes](https://www.sciencedirect.com/science/article/pii/0550321391902926) *predicted* the *virtual* number $n\_d$ of rational curves of degree $d$ on a quintic threefold *for any* $d\geqslant 1$. The numbers $n\_d$ are defined by the 'multiple cover' formula
$$\sum\_{d=1}^\infty N\_d q^d=\sum\_{d=1}^\infty \sum\_{k=1}^\infty \frac{n\_d}{k^3}q^{kd}$$
in terms of other numbers $N\_d$, the *genus $0$ Gromov-Witten invariants* (which, roughly speaking, count *maps* $f$ – not just embedded curves – from $\mathbf{P}^1$ to the quintic threefold with $f\_\ast[\mathbf{P}^1]=d$). The $N\_d$ need not be integers, and are in general only rational numbers – e.g. one has $N\_2=4876875/8$. How the physicists used a change of variables, the 'mirror map', to equate a generating series of the $n\_d$ with an explicitly calculable series associated to the 'mirror' of the quintic is best described elsewhere (e.g. in [this](http://this) book, in various levels of details, or the Perutz article linked in the comments). 


In any case, I would like to point out that it is a subtle question if the numbers $n\_d$ are *enumerative*, that is, $n\_d$ is the actual number of rational curves of degree $d$ in the quintic\*. For example, the $n\_d$ are deformation-invariant (because the $N\_d$ are) and so independent of the quintic $X$, but the actual number is certainly not deformation-invariant. Think of the classical case $d=1$ (lines), $n\_1=2875$ computed by [Schubert](https://link.springer.com/article/10.1007/BF01443569). If you take the equation of the quintic $X$ to be $x\_0^5+\cdots+x\_3^5=0$, then the number of lines on $X$ is not even finite, in fact the moduli space of lines on $X$ has dimension $1$. However, if $X$ is 'generic', then a fairly classical and elementary argument shows that the moduli space of lines is $0$-dimensional and reduced, and the number of points can easily be computed by using the intersection theory of the Grassmannian $\mathrm{Gr}(2,5)$.


For $d>1$ the analysis of the moduli space of rational curves of degree $d$ on $X$ and the intersection theory calculations become much more complicated. For $d=2$ this was carried out by [Katz](http://www.numdam.org/article/CM_1986__60_2_151_0.pdf), while the $d=3$ case was handled by [Ellingsrud and Strømme](https://www.jstor.org/stable/24491856?seq=1#page_scan_tab_contents). Curiously, the number first computed by Ellingsrud and Strømme in $1991$ was $2682549425$, in disagreement with the number $n\_3=317206375$ computed by Candelas, De La Ossa, Green, Parkes. Ellingsrud and Strømme then found an error in their computer program – *in a sense, string theory served as a debugger*. (The history of the $d=3$ case is described in great detail in [this article by P. Galison](https://galison.scholar.harvard.edu/publications/mirror-symmetry-persons-values-and-objects).)


\*Indeed, the number $n\_{10}$ [is not enumerative](https://arxiv.org/abs/math/0412002).


