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More generally, why would we (mathematicians, not physicists) want to know anything about any Gromov-Witten invariants at all? For me, they're just interesting invariants of varieties/manifolds. The structure of Gromov-Witten invariants is quite rich and interesting on its own.

Perhaps your computation can tell you something interesting about another variety, via mirror symmetry. And yes, it is often the case that Gromov-Witten invariants make contact with integrable systems, but I seem to have the impression that most of the time, the applications go in the direction of integrable systems helping our understanding of Gromov-Witten theory --- e.g. Witten's conjecture, Virasoro conjecture --- rather than the other way around.

Another thing is that, since Gromov-Witten invariants are not really enumerative invariants, it can sometimes be interesting to try to relate them to "actual" enumerative invariants. Moreover there are various questions about the relationship between Gromov-Witten invariants and other "enumerative" invariants like Donaldson-Thomas invariants. There are sometimes also connections with things like matrix models. For example, this paper of Okounkov-Pandharipande relates Hurwitz numbers ("actual" enumerative invariants), the Gromov-Witten theory of P^1, and matrix models. Okounkov and Pandharipande have written many papers along the lines of "the quantum cohomology of X is blah" or "the Gromov-Witten theory of X is blah".

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More generally, why would we (mathematicians, not physicists) want to know anything about any Gromov-Witten invariants at all? For me, they're just interesting invariants of varieties/manifolds. The structure of Gromov-Witten invariants is quite rich and interesting on its own.

But perhaps

Perhaps your computation can tell you something interesting about another variety, via mirror symmetry. And yes, it is often the case that Gromov-Witten invariants make contact with integrable systems, but I seem to have the impression that most of the time, the applications go in the direction of integrable systems helping our understanding of Gromov-Witten theory --- e.g. Witten's conjecture, Virasoro conjecture --- rather than the other way around.

Another thing is that, since Gromov-Witten invariants are not really enumerative invariants, it can sometimes be interesting to try to relate them to "actual" enumerative invariants. Moreover there are various questions about the relationship between Gromov-Witten invariants and other "enumerative" invariants like Donaldson-Thomas invariants. There are sometimes also connections with things like matrix models. For example, this paper of Okounkov-Pandharipande relates Hurwitz numbers ("actual" enumerative invariants), the Gromov-Witten theory of P^1, and matrix models.

1

More generally, why would we (mathematicians, not physicists) want to know anything about any Gromov-Witten invariants at all? For me, they're just interesting invariants of varieties/manifolds. The structure of Gromov-Witten invariants is quite rich and interesting on its own.

But perhaps your computation can tell you something interesting about another variety, via mirror symmetry. And yes, it is often the case that Gromov-Witten invariants make contact with integrable systems, but I seem to have the impression that most of the time, the applications go in the direction of integrable systems helping our understanding of Gromov-Witten theory --- e.g. Witten's conjecture, Virasoro conjecture --- rather than the other way around.