Genus-zero Gromov-Witten invariants $N_d$ of the projective plane

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Numbers

$d$

$N_d$

620

87304

26312976

14616808192

13525751027392

19385778269260800

10:

40739017561997799680

11:

120278021410937387514880

12:

482113680618029292368686080

13:

2551154673732472157928033617920

14:

17410560213476464590484763013222400

Definition

For $d\geq 1$, $N_d=\langle [\mathrm{pt}],\ldots,[\mathrm{pt}]\rangle_{0,d}^{\mathbb P^2}$ is the genus-zero Gromov-Witten invariant with $3d-1$ point insertions [3].

Parameters

$d$

— degree ($d\geq 1$)

Formulas

(1)

$N_1=1$, and for $d\geq 2$, $N_d=\sum_{a+b=d,\;a,b\geq 1}N_aN_ba^2b\left(b\binom{3d-4}{3a-2} -a\binom{3d-4}{3a-1}\right)$ [1].

Comments

(2)

Since $\mathbb P^2$ is homogeneous, $N_d$ is the enumerative count [4] of degree-$d$ rational curves in $\mathbb P^2$ through $3d-1$ points in general position [1]. The condition count $3d-1$ is the dimension of that family.

(3)

The table uses the usual complex enumerative convention. It does not store Welschinger's signed real counts, which are different invariants.

Programs

(P1)

Python

from math import comb

def plane_gromov_witten(limit):
    values = {1: 1}
    for d in range(2, limit + 1):
        total = 0
        for a in range(1, d):
            b = d - a
            total += (values[a] * values[b] * a * a * b
                      * (b * comb(3 * d - 4, 3 * a - 2)
                         - a * comb(3 * d - 4, 3 * a - 1)))
        values[d] = total
    return values

plane_gromov_witten(15)[15]

References

[1]

M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Communications in Mathematical Physics 164 (1994), 525-562. (arXiv) (doi)

[2]

P. Di Francesco and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, Progress in Mathematics 129, Birkhauser Boston, 1995, 81-148. (arXiv) (doi)

Links

[3]

Wikipedia: Gromov-Witten invariant

[4]

Wikipedia: Enumerative geometry

[5]

OEIS A013587

Data properties

Entries are of type: integer

How well the digits are known: exact

Table is complete: no (it holds every degree $1\leq d\leq 14$; by then the values have 35 digits, so the table stops with the small-degree counts rather than reproducing the 169-term OEIS b-file)

How they were obtained:

The generator evaluates Formula (1) in exact integer arithmetic. The values were compared with OEIS A013587 [5] for every entry, and the entries with $d\leq 12$ were compared with the table in [2].