# Did physicists correct an error of mathematicians in counting twisted cubics in the quintic?

One problem in enumerative geometry consists in counting the number of rational curves of degree $$d$$ in the plane going through $$n$$ general points. If $$n = 3d-1$$, this number, denoted $$N_d$$, is finite and the problem is to find a formula for $$N_d$$.
Now, one can ask the same question but for rational curves not on the plane but on a quintic hypersurface given by the equation $$x_1^5 + x_2^5 + x_3^5 + x_4^5 + x_5^5 = 0$$ in projective $$4$$-space. In this last case, a classical result from the XIXth century gives $$N_1 = 2875$$, i.e. there are $$2875$$ lines on the quintic. The next number $$N_2$$ counts the number of conics in the quintic and was found around 1980. The number of twisted cubics, $$N_3 = 317 206 375$$, was the result of a complicated computer program.
I remember a mathematican telling me there was a bug in the program and mathematicians were not aware of it initially. Actually, physicists noticed an error in the mathematicians's computation due to some discrepancy with their own result on the same problem calculated with very different methods (not the result of a computer program). As a result, physicists called the mathematicians's attention to a possible bug in their computer program. As it happens, the physicists were right and a bug was hidden in the computer program of the mathematicians.
Can someone confirm this story or correct it? If the story is true, please provide details and references.

Indeed they did, the OP story is told by Yau in The Shape of Inner Space, pp.169-70. Calculating the number of twisted cubics in the quintic hypersurface was a big early coup for mirror symmetry that attracted mathematicians' attention to string theory and revitalized enumerative geometry. In A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory (1991) Candelas, De La Ossa, Green and Parkes managed to compute what was beyond the means of mathematicians at the time. As Katz remarked in On the finiteness of rational curves on quintic threefolds (1986):

"It would be very interesting to know what the number of twisted cubics on a generic quintic is. A more detailed understanding of the moduli space of twisted cubics is necessary. In [9], we see that this number is divisible by 5, by studying the degeneration of the quintic to a union of 5 hyperplanes, along the lines of [8]."

In fact, COGP computed the number of rational curves of degree up to $$10$$. The computer error story began at the 1991 MSRI conference, where Candelas reported their results which left most mathematicians skeptical. In addition to unfamiliarity and non-rigorous nature of the new methods, the number $$317,206,375$$ did not match the number $$2,682,549,425$$ obtained by Norwegian mathematicians Ellingsrud and Strømme using a computer.

However, a month later they noticed an error in the program, and in The Number of Twisted Cubic Curves on the General Quintic threefold (1995) rederived the COGP result for twisted cubics theoretically. This moved the attitudes. Morrison, one of the most vocal critics of COGP at MSRI, became one of the leading lights in mirror symmetry and mathematical string theory. Kontsevich won the Fields medal in 1998 in part for confirming the degree four count $$242,467,530,000$$ in 1995, and turning semi-physical heuristics of mirror symmetry into a system of precise formulations that could be rigorously proved.

Here is Yau's account:

"Fortuitously, at about this time—May 1991, to be exact — I had organized a conference at the Mathematical Sciences Research Institute (MSRI) at Berkeley for mathematicians and physicists to talk about mirror symmetry... Candelas discussed his result for the Schubert problem there, but his number disagreed with the number obtained through ostensibly more rigorous techniques by the Norwegian mathematicians Geir Ellingsrud and Stein Arild Strømme, who had arrived at 2,682,549,425. The algebraic geometers in the crowd tended to be arrogant, assuming the physicists must have made the mistake. For one thing, explains University of Kaiserslautern mathematician Andreas Gathmann, “mathematicians didn’t understand what the physicists were doing, because they [the physicists] were using completely different methods — methods that didn’t exist in math and didn’t always appear to be justified.”

Candelas and Greene worried that they might have made an error, but they couldn’t see where they had gone wrong. I spoke with both of them at the time, especially Greene, wondering whether something might have gone awry during the process of integrating over an infinite-dimensional space that then has to be reduced to finite dimensions. Choices have to be made in the course of doing that, and none of the options are perfect. While that made Candelas and Greene somewhat uneasy, we couldn’t find any flaws in their reasoning, which was based on physics rather than an outright mathematical proof. Moreover, they remained confident about mirror symmetry in general, despite the mathematicians’ criticism.

This all became clear a month or so later, when Ellingsrud and Strømme found an error in their computer program. Upon rectifying it, they got the same answer that Candelas et al. had. The Norwegian mathematicians showed great integrity in rerunning their program, checking their results, and publicizing their mistake. Many people in the same position try to conceal a mistake for as long as possible, but Ellingsrud and Strømme did the opposite, promptly informing the community of their error and subsequent correction".

• Do you know if the error was a subtle bug in the software used or a flaw in the reasoning which was encoded in their computer program? I skimmed (quickly) through Ellingsrud and Stromme's 1995 article and I found no mention of their initial mistake. Nov 9, 2020 at 20:27
• @AnsonīBōdo Unfortunately, no. I am not even sure how exactly Ellingsrud and Strømme "promptly informed the community". I also looked through their paper and searched, but could not find a record of it other than Yau's report. Nov 9, 2020 at 22:17