Genus-zero Gopakumar–Vafa invariants of Calabi–Yau complete intersections in projective space

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Numbers

$\ell$

$d$

$n_{\ell,d}$

$(5)$

2875

$(5)$

609250

$(5)$

317206375

$(5)$

242467530000

$(5)$

229305888887625

$(5)$

248249742118022000

$(5)$

295091050570845659250

$(5)$

375632160937476603550000

$(5)$

503840510416985243645106250

$(5)$

10:

704288164978454686113488249750

$(5)$

11:

1017913203569692432490203659468875

$(5)$

12:

1512323901934139334751675234074638000

$(5)$

13:

2299488568136266648325160104772265542625

$(5)$

14:

3565959228158001564810294084668822024070250

$(5)$

15:

5624656824668483274179483938371579753751395250

$(5)$

16:

9004003639871055462831535610291411200360685606000

$(5)$

17:

14602074714589033874568888115959699651605558686799250

$(5)$

18:

23954445228532694121482634657728114956109652255216482000

$(5)$

19:

39701666985451876233836105884497728824100003703180307231625

$(5)$

20:

66408603312404471392397268104340892583652834904833089314920000

$(2,4)$

1280

$(2,4)$

92288

$(2,4)$

15655168

$(2,4)$

3883902528

$(2,4)$

1190923282176

$(2,4)$

417874605342336

$(2,4)$

160964588281789696

$(2,4)$

66392895625625639488

$(2,4)$

28855060316616488359936

$(2,4)$

10:

13069047760169269024822656

$(2,4)$

11:

6120237475407486434666617088

$(2,4)$

12:

2946240164493101121609807042880

$(2,4)$

13:

1451521723048555184993212271677184

$(2,4)$

14:

729359127416597618928975536123540608

$(2,4)$

15:

372767405871565200223492775287001701120

$(2,4)$

16:

193356406784208385213827499686874510015040

$(2,4)$

17:

101606751091857407788281087029439219147286272

$(2,4)$

18:

54011118222274393294176294370838415376632364032

$(2,4)$

19:

29006721031375167678423543580812641523573783433472

$(2,4)$

20:

15722105045068782170402865503903134307265397121634496

$(3,3)$

1053

$(3,3)$

52812

$(3,3)$

6424326

$(3,3)$

1139448384

$(3,3)$

249787892583

$(3,3)$

62660964509532

$(3,3)$

17256453900822009

$(3,3)$

5088842568426162960

$(3,3)$

1581250717976557887945

$(3,3)$

10:

512045241907209106828608

$(3,3)$

11:

171443844967715602251281694

$(3,3)$

12:

59008426658919570084145507296

$(3,3)$

13:

20785681827642410211701991235896

$(3,3)$

14:

7467568617384291804021081709713036

$(3,3)$

15:

2728818932354545979223930630299992371

$(3,3)$

16:

1012036150577299374980757071904131847504

$(3,3)$

17:

380243831850088432217116550564067293704578

$(3,3)$

18:

144519433563613558831955702896560953425168536

$(3,3)$

19:

55494042563903494815332991175873000067100313938

$(3,3)$

20:

21506221523280488684717732835443600864733944700672

$(2,2,3)$

720

$(2,2,3)$

22428

$(2,2,3)$

1611504

$(2,2,3)$

168199200

$(2,2,3)$

21676931712

$(2,2,3)$

3195557904564

$(2,2,3)$

517064870788848

$(2,2,3)$

89580965599606752

$(2,2,3)$

16352303769375910848

$(2,2,3)$

10:

3110686153486233022944

$(2,2,3)$

11:

611833306260743839686384

$(2,2,3)$

12:

123704510635152339063953760

$(2,2,3)$

13:

25597344552035868430770363264

$(2,2,3)$

14:

5402179701272176605078804458676

$(2,2,3)$

15:

1159643324896738468504628336070048

$(2,2,3)$

16:

252642739904612108584900275742488288

$(2,2,3)$

17:

55761633516219958401057559179207023472

$(2,2,3)$

18:

12449836369698886589723738565198615892944

$(2,2,3)$

19:

2808338626888242040699317356197140206652816

$(2,2,3)$

20:

639344002217231077328235520944763941384329472

$(2,2,2,2)$

512

$(2,2,2,2)$

9728

$(2,2,2,2)$

416256

$(2,2,2,2)$

25703936

$(2,2,2,2)$

1957983744

$(2,2,2,2)$

170535923200

$(2,2,2,2)$

16300354777600

$(2,2,2,2)$

1668063096387072

$(2,2,2,2)$

179845756064329728

$(2,2,2,2)$

10:

20206497983891554816

$(2,2,2,2)$

11:

2347339059011866069504

$(2,2,2,2)$

12:

280306600807533481197056

$(2,2,2,2)$

13:

34256798893452529348216320

$(2,2,2,2)$

14:

4269972051694179948749436416

$(2,2,2,2)$

15:

541358921805547172080502619648

$(2,2,2,2)$

16:

69658426344401004665507307501056

$(2,2,2,2)$

17:

9080497284788411740962012219340288

$(2,2,2,2)$

18:

1197420141818887534508408000477679616

$(2,2,2,2)$

19:

159530114843856823677943126671820601856

$(2,2,2,2)$

20:

21450608635104560649049233453201791676928

Definition

The genus-zero Gopakumar–Vafa invariant $n_{\ell,d}$ of a smooth Calabi–Yau threefold complete intersection $X_\ell$ is defined by $N_{\ell,d}=\sum_{k\mid d}n_{\ell,d/k}/k^3$, where $N_{\ell,d}$ is its connected, unmarked genus-zero Gromov–Witten invariant [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 $K_\ell(q)=\dfrac{\kappa}{(1-Cx)F(x)^2}\left(\dfrac{q}{x}\dfrac{dx}{dq}\right)^3=\kappa+\sum_{d\geq1}n_{\ell,d}\dfrac{d^3q^d}{1-q^d}$. Here $\kappa=\prod_i\ell_i$ and $C=\prod_i\ell_i^{\ell_i}$. The mirror theorem supplies this enumerative expansion [2] [3].

(3)

The Gromov–Witten 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. Thus $K_\ell(q)=\kappa+\sum_{d\geq1}d^3N_{\ell,d}q^d$ [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 $(2,2,2,2)$ [5]. 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 $H=c_1(\mathcal O(1)|_{X_\ell})$. The underlying stable-map invariant is $N_{\ell,d}=\int_{[\overline{\mathcal M}_{0,0}(X_\ell,d)]^{\mathrm{vir}}}1$; there are no marked points, insertions or descendants. The integers $n_{\ell,d}$ remove the rational multiple-cover contributions. They are also called genus-zero instanton numbers or BPS invariants, following their interpretation in terms of BPS states [1]. They are not asserted to count embedded rational curves literally in every degree.

(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 geometric Yukawa series is $\kappa=\int_{X_\ell}H^3$, rather than $1$. Degree zero contributes this classical term and is not a row of the table. The normalized differential-operator instanton numbers in [6] must be multiplied by $\kappa$ to give the convention used here.

(9)

Integrality of the Gopakumar–Vafa invariants for Calabi–Yau threefolds is a theorem [4]. In degree one the invariant is the number of lines on a general member; in particular, the quintic has $2875$ lines [5].

Programs

(P1)

Macaulay2

needsPackage "EnumerationCurves";
multidegree = {5};
-- The degree-two invariant removes the double covers of lines.
rationalCurve(2, multidegree) - rationalCurve(1, multidegree)/8

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)

[4]

E.-N. Ionel and T. H. Parker, The Gopakumar-Vafa formula for symplectic manifolds, Annals of Mathematics 187 (2018), 1–64. (doi) (MR)

Links

[5]

Macaulay2 EnumerationCurves: rationalCurve

[6]

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

Similar tables

Genus-zero Gromov–Witten invariants of Calabi–Yau complete intersections in projective space — gives the rational stable-map invariants of the same five targets, related by the multiple-cover formula

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

Data properties

Entries are of type: integer

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 covering low-degree enumerative controls and successive prime and composite degrees across every target while keeping exact values readable. The longest integer has $62$ digits. The attached argument-selection note explains the scope and bounds)

How they were obtained:

Computed using exact rational arithmetic for the geometric Yukawa series, followed by triangular Lambert-series extraction. Every stored value is an exact integer. All 100 values agree with a separately implemented second-logarithmic-period calculation using nilpotent-parameter products, Lagrange inversion and Möbius inversion. Series truncations through orders 24 and 28 agree through degree 20. These mirror calculations share the underlying differential system; their agreement checks the implementations rather than proving mirror symmetry.

Independent geometric controls comprise the 18 instanton values documented in [5]: degrees 1 through 3 for all five targets, and degree 4 for $(5)$, $(2,4)$ and $(3,3)$. The underlying rational series also agrees with its 17 documented stable-map outputs. A separate Grassmannian intersection calculation reproduces the line counts of all five targets, after a control giving 27 lines on a cubic surface. Macaulay2 was unavailable in the build environment, so its documented outputs were compared without rerunning that package. Exact coefficient comparisons check the template and coordinate correspondence. The attached checks and numerical evidence also record comparisons with the companion table and with the values read back from this draft.