Genus-zero Gromov–Witten invariants of Calabi–Yau complete intersections in projective space

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Numbers

$\ell$

$d$

$N_{\ell,d}$

$(5)$

2875

$(5)$

4876875/8

$(5)$

8564575000/27

$(5)$

15517926796875/64

$(5)$

229305888887648

$(5)$

248249742157695375

$(5)$

101216230345800061125625/343

$(5)$

192323666400003538944396875/512

$(5)$

367299732093982242625847031250/729

$(5)$

10:

704288164978454714776724365580

$(5)$

11:

1354842473951260627644461070753075500/1331

$(5)$

12:

108887320939258032104354865179373274375/72

$(5)$

13:

5051976384195377826370376750184667397150000/2197

$(5)$

14:

9784992122065556293839548184561593434114765625/2744

$(5)$

15:

151865734266049048402846066336032882657245075975/27

$(5)$

16:

36880398908911843175757970052077286676680907186572875/4096

$(5)$

17:

71739993072775923425756947313710004388338109828244718125/4913

$(5)$

18:

287453342742392329457791615893493140239051638146802196875/12

$(5)$

19:

272313733853214419087881850261769922004501925400113727301718750/6859

$(5)$

20:

132817206624808942784794536208681961239346914423344876690411875/2

$(2,4)$

1280

$(2,4)$

92448

$(2,4)$

422690816/27

$(2,4)$

3883914084

$(2,4)$

29773082054656/25

$(2,4)$

417874607302656

$(2,4)$

55210853780653867008/343

$(2,4)$

132785791252222257497/2

$(2,4)$

21035338970813420437084160/729

$(2,4)$

10:

1633630970021177236279208448/125

$(2,4)$

11:

8146036079767364444541267345408/1331

$(2,4)$

12:

26516161480437910564598491235584/9

$(2,4)$

13:

3188993225537675741430087360874774528/2197

$(2,4)$

14:

250170180703892983299539965612956254208/343

$(2,4)$

15:

1258089994816532550754288116742496574242816/3375

$(2,4)$

16:

3093702508547334163421372780781244382498137/16

$(2,4)$

17:

499193968114295444463824980575634883670617455616/4913

$(2,4)$

18:

486100064000469539647586681799488733874776448768/9

$(2,4)$

19:

198957099554202275106307085420793908210192580570185728/6859

$(2,4)$

20:

1965263130633597771300358188192095659660821798623115584/125

$(3,3)$

1053

$(3,3)$

423549/8

$(3,3)$

6424365

$(3,3)$

72925120125/64

$(3,3)$

31223486573928/125

$(3,3)$

501287722516269/8

$(3,3)$

5918963687981950140/343

$(3,3)$

2605487395107120555645/512

$(3,3)$

14231256461789023132960/9

$(3,3)$

10:

64005655238405041289450553/125

$(3,3)$

11:

228191757652029466596455935767/1331

$(3,3)$

12:

3776539306170852986675735897901/64

$(3,3)$

13:

3512780228871567325777636518866505/169

$(3,3)$

14:

731821724503660597005457567836962265/98

$(3,3)$

15:

68220473308863649480598265988786179873/25

$(3,3)$

16:

4145300072764618239923786453914431167932029/4096

$(3,3)$

17:

1868137945879484467482693612921262613970592767/4913

$(3,3)$

18:

144519433563613558831955900552903021271397660

$(3,3)$

19:

380633637945814070938368986475312907460241053301795/6859

$(3,3)$

20:

172049772186243909477741863195594048825111960845677/8

$(2,2,3)$

720

$(2,2,3)$

22518

$(2,2,3)$

4834592/3

$(2,2,3)$

672808059/4

$(2,2,3)$

541923292944/25

$(2,2,3)$

3195558106836

$(2,2,3)$

177353250680575584/343

$(2,2,3)$

2866590899860224123/32

$(2,2,3)$

1324536605319453613280/81

$(2,2,3)$

10:

388835769186117829948518/125

$(2,2,3)$

11:

814350130633050050622577824/1331

$(2,2,3)$

12:

247409021270305477029893429/2

$(2,2,3)$

13:

56237365980822802942402488091728/2197

$(2,2,3)$

14:

1852947637536356597711186264420244/343

$(2,2,3)$

15:

144955415612092308563078642366532864/125

$(2,2,3)$

16:

64676541415580699800601061489937225851/256

$(2,2,3)$

17:

273956905465188655624395788247444106318656/4913

$(2,2,3)$

18:

112048527327289979307532043429593276971812/9

$(2,2,3)$

19:

19262394641826452157156617746156184677431665664/6859

$(2,2,3)$

20:

319672001108615538664117954890266563751752507059/500

$(2,2,2,2)$

512

$(2,2,2,2)$

9792

$(2,2,2,2)$

11239424/27

$(2,2,2,2)$

25705160

$(2,2,2,2)$

244747968512/125

$(2,2,2,2)$

511607926784/3

$(2,2,2,2)$

5591021688717312/343

$(2,2,2,2)$

1668063099600217

$(2,2,2,2)$

131107556170907611136/729

$(2,2,2,2)$

10:

2525812248017037857792/125

$(2,2,2,2)$

11:

3124308287544793738510336/1331

$(2,2,2,2)$

12:

7568278221803979576942080/27

$(2,2,2,2)$

13:

75262187168915206978031255552/2197

$(2,2,2,2)$

14:

1464600413731104421298767790080/343

$(2,2,2,2)$

15:

1827086361093721705771941100519424/3375

$(2,2,2,2)$

16:

557267410755208038992121559608665/8

$(2,2,2,2)$

17:

44612483160165466883346366033618835456/4913

$(2,2,2,2)$

18:

96991031487329890297001986830451314176/81

$(2,2,2,2)$

19:

1094217057714013953607011905842017508130816/6859

$(2,2,2,2)$

20:

536265215877614016226293981636245223010432/25

Definition

The connected, unmarked genus-zero Gromov–Witten invariant of a smooth Calabi–Yau threefold complete intersection $X_\ell$ is $N_{\ell,d}=\int_{[\overline{\mathcal M}_{0,0}(X_\ell,d)]^{\mathrm{vir}}}1$ [1].

Parameters

$\ell$

— multidegree (a nondecreasing tuple $(\ell_1,\ldots,\ell_r)$ of integers at least $2$, with $\sum_i(\ell_i-1)=4$)

$d$

— curve degree ($d\geq1$)

Formulas

(1)

Put $H_m=\sum_{j=1}^m1/j$, with $H_0=0$. Then $a_m=\prod_i(\ell_i m)!/(m!)^{r+4}$, $F(x)=\sum_{m\geq0}a_mx^m$, $G(x)=\sum_{m\geq1}a_m\bigl(\sum_i\ell_iH_{\ell_i m}-(r+4)H_m\bigr)x^m$, and $q=x\exp(G/F)$ [2].

(2)

For the formal inverse $x=x(q)$, the geometric Yukawa series is $K_\ell(q)=\dfrac{\kappa}{(1-Cx)F(x)^2}\left(\dfrac{q}{x}\dfrac{dx}{dq}\right)^3=\kappa+\sum_{d\geq1}d^3N_{\ell,d}q^d$. The mirror theorem identifies these coefficients with the stable-map invariants [2] [3].

(3)

The genus-zero Gopakumar–Vafa invariants satisfy $N_{\ell,d}=\sum_{k\mid d}n_{\ell,d/k}/k^3$ and $n_{\ell,d}=\sum_{k\mid d}\mu(k)N_{\ell,d/k}/k^3$, where $\mu$ is the Möbius function. Equivalently, $K_\ell(q)=\kappa+\sum_{d\geq1}n_{\ell,d}d^3q^d/(1-q^d)$ [1].

Comments

(4)

Here $X_\ell\subset\mathbb P^{r+3}_{\mathbb C}$ has multidegree $\ell=(\ell_1,\ldots,\ell_r)$, with $r$ the number of defining hypersurfaces. The five multidegrees are $(5)$, $(2,4)$, $(3,3)$, $(2,2,3)$ and $(2,2,2,2)$ [4]. They exhaust the partitions of $\sum_i(\ell_i-1)=4$, after linear equations have been eliminated. The quintic has multidegree $(5)$. The invariants are unchanged under deformation through smooth complete intersections of fixed multidegree.

(5)

The degree is $d=\int_\beta H$ for the primitive hyperplane class $H=c_1(\mathcal O(1)|_{X_\ell})$. There are no marked points, insertions or descendants. The virtual dimension is zero, and multiple covers contribute rational weights. Thus $N_{\ell,d}$ need not be an integer or the literal number of embedded rational curves. The genus-zero Gopakumar–Vafa invariants $n_{\ell,d}$ remove these multiple-cover contributions by Formula (3) [1].

(6)

Adjunction gives $c_1(T_{X_\ell})=(r+4-\sum_i\ell_i)H=0$ and the classical intersection number is $\kappa=\int_{X_\ell}H^3=\prod_i\ell_i$. Chern classes and Euler characteristics use the same target multidegree and hyperplane polarization.

(7)

The formal variable $x$ in Formula (1) is related to the hypergeometric argument by $z=Cx$, where $C=\prod_i\ell_i^{\ell_i}$. In the normalized period table and the canonical-coordinate table, the multidegrees $(5)$, $(2,4)$, $(3,3)$, $(2,2,3)$ and $(2,2,2,2)$ correspond respectively to template keys $5$, $2.2.4$, $3.3$, $2.2.3$ and $2.2.2.2$. Their scale constants are respectively $3125$, $1024$, $729$, $432$ and $256$. The functions satisfy $F(x)=F_A(Cx)$ and $q(x)=q_A(Cx)$, with the integral coordinate $q_A=Q_A/C$; $q/x\to1$. No complex-structure fibre is selected for these curve invariants.

(8)

The constant term of the Yukawa series is $\kappa$, rather than $1$. Degree zero contributes this classical term and is not a row of the table. The differential-operator convention in [5] instead normalizes the constant term to $1$.

Programs

(P1)

Macaulay2

needsPackage "EnumerationCurves";
d = 2;
multidegree = {5};
rationalCurve(d, multidegree)

References

[1]

Y.-C. Chou and Y.-P. Lee, Quantum K-invariants and Gopakumar–Vafa invariants II. Calabi-Yau threefolds at genus zero, Section 1.1. (arXiv)

[2]

A. B. Givental, Equivariant Gromov-Witten invariants, introductory theorem and Section 11. (arXiv)

[3]

A. Givental, A mirror theorem for toric complete intersections, Theorem 0.1 and its Calabi–Yau specialization. (arXiv)

Links

[4]

Macaulay2 EnumerationCurves: rationalCurve

[5]

Calabi–Yau differential-operator database: Frobenius basis and normalized instanton numbers

Similar tables

Normalized holomorphic periods of hypergeometric Calabi–Yau families — gives the normalized holomorphic periods whose factorial series enter the mirror construction, with $z=Cx$

Canonical coordinates (mirror maps) of hypergeometric Calabi–Yau families — gives the corresponding canonical coordinates; its integral convention agrees with $q(x)=q_A(Cx)$

Chern classes of smooth complete intersections in $\mathbb{P}^n$ — gives Chern classes of the same targets with the same hyperplane polarization

Euler characteristics of smooth complete intersections in $\mathbb{P}^n$ — gives topological Euler characteristics of the same targets

Genus-zero Gromov-Witten invariants $N_d$ of the projective plane — counts rational curves on $\mathbb P^2$ with $3d-1$ point insertions, whereas these threefold invariants have no insertions

Frobenius polynomials of hypergeometric Calabi–Yau motives — gives local arithmetic of the associated mirror differential systems at specified smooth rational fibres

Genus-zero Gopakumar–Vafa invariants of Calabi–Yau complete intersections in projective space — stores the genus-zero Gopakumar–Vafa invariants $n_{\ell,d}$ of the same five targets. They remove multiple-cover contributions by $N_{\ell,d}=\sum_{k\mid d}n_{\ell,d/k}/k^3$, or equivalently $n_{\ell,d}=\sum_{k\mid d}\mu(k)N_{\ell,d/k}/k^3$, where $\mu$ is the Möbius function

Data properties

Entries are of type: rational number

How well the digits are known: exact

Table is complete: no (it holds all five ordinary-projective Calabi–Yau threefold complete-intersection types, $(5)$, $(2,4)$, $(3,3)$, $(2,2,3)$ and $(2,2,2,2)$, in every curve degree $1\leq d\leq20$. The geometric class is exhaustive; degree $20$ is a noncanonical practical cutoff providing low-degree controls and successive prime and composite degrees across every target while keeping exact values readable. The longest rational has $68$ characters. The attached argument-selection note explains the scope and bounds)

How they were obtained:

Computed in exact rational arithmetic from Formula (2). All 100 values agree with a separately implemented second-logarithmic-period calculation using nilpotent-parameter products and Lagrange inversion. Recomputations through orders 24 and 28 agree through degree 20. These two mirror calculations share the underlying differential system; their agreement is an implementation check, not an independent proof of mirror symmetry.

Independent geometric controls comprise the 17 localization outputs documented in [4]: degrees 1 through 3 for all five targets and degree 4 for $(5)$ and $(2,4)$. A separate Grassmannian intersection calculation reproduces the degree-one virtual counts for all five types. Macaulay2 was unavailable in the build environment, so its documentation outputs were compared without rerunning that package. Exact coefficient comparisons check the template and coordinate correspondence, and Lambert-series extraction yields integral coefficients throughout. The attached checks, source notes and numerical evidence record the methods and results.