Calabi-Yau differential operator database v.3.0

1

New Number: 2.2 | AESZ: 15 | Superseeker: 21 15894 | Hash: c8053e0e9c05ef468263fafd5e3fc764

Degree: 2

$\theta^4-3 x(3\theta+1)(3\theta+2)(7\theta^2+7\theta+2)-2^{3} 3^{2} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 900, 94080, 11988900, ...
--> OEIS
Normalized instanton numbers (n₀=1): 21, 480, 15894, 894075, 58703151, ... ; Common denominator:...

Discriminant

$-(27z+1)(216z-1)$

Local exponents

$-\frac{ 1}{ 27}$ $0$ $\frac{ 1}{ 216}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 3}$
$1$ $0$ $1$ $\frac{ 2}{ 3}$
$1$ $0$ $1$ $\frac{ 4}{ 3}$
$2$ $0$ $2$ $\frac{ 5}{ 3}$

\(-\frac{ 1}{ 27}\)	\(0\)	\(\frac{ 1}{ 216}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 2}{ 3}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 4}{ 3}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 5}{ 3}\)

Note:

Hadamard product $B\ast a$.

A-Incarnation: diagonal of (3,3)-intersection in $P^2 \times P^2$

	Gopakumar-Vafa invariants
g=0	,...
g=1	,...
g=2	,...

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-162\)	\(72\)	\(18\)	\(9\)

2

New Number: 2.35 | AESZ: ~67 | Superseeker: 480 -16034720 | Hash: f06ee3928cd6d738db065f3f83d12160

Degree: 2

$\theta^4-2^{4} 3 x(2\theta+1)^2(72\theta^2+72\theta+31)+2^{12} 3^{6} x^{2}(2\theta+1)^2(2\theta+3)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 1488, 5351184, 24363091200, 123873273392400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 480, -226968, -16034720, 10943202744, -4352645747040, ... ; Common denominator:...

Discriminant

$(6912z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 6912}$ $\infty$
$0$ $0$ $\frac{ 1}{ 2}$
$0$ $\frac{ 1}{ 6}$ $\frac{ 1}{ 2}$
$0$ $\frac{ 5}{ 6}$ $\frac{ 3}{ 2}$
$0$ $1$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 6912}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(\frac{ 1}{ 6}\)	\(\frac{ 1}{ 2}\)
\(0\)	\(\frac{ 5}{ 6}\)	\(\frac{ 3}{ 2}\)
\(0\)	\(1\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "2.35" from ...

	Gopakumar-Vafa invariants
g=0	,...
g=1	,...
g=2	,...

3

New Number: 4.52 | AESZ: 258 | Superseeker: 480 4215904 | Hash: bfb9f01124fd9980817cbf1b50f789c3

Degree: 4

$\theta^4-2^{4} x\left(16\theta^4+224\theta^3+156\theta^2+44\theta+5\right)-2^{14} x^{2}\left(48\theta^4+48\theta^3-120\theta^2-66\theta-11\right)-2^{22} x^{3}\left(16\theta^4-192\theta^3-156\theta^2-48\theta-5\right)+2^{32} x^{4}\left((2\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 80, 24336, 11398400, 6632189200, ...
--> OEIS
Normalized instanton numbers (n₀=1): 480, -16536, 4215904, -242723592, 151800032928, ... ; Common denominator:...

Discriminant

$(1024z-1)(256z-1)(1+512z)^2$

Local exponents

$-\frac{ 1}{ 512}$ $0$ $\frac{ 1}{ 1024}$ $\frac{ 1}{ 256}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $\frac{ 1}{ 2}$
$3$ $0$ $1$ $1$ $\frac{ 1}{ 2}$
$4$ $0$ $2$ $2$ $\frac{ 1}{ 2}$

\(-\frac{ 1}{ 512}\)	\(0\)	\(\frac{ 1}{ 1024}\)	\(\frac{ 1}{ 256}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(\frac{ 1}{ 2}\)

Note:

Sporadic Operator.

\(4+324\lambda\)	\(-\frac{ 3}{ 2}-162\lambda\)	\(\frac{ 3}{ 4}+81\lambda\)	\(-\frac{ 222}{ 1223}-54\lambda\)
\(9\)	\(-\frac{ 7}{ 2}\)	\(\frac{ 9}{ 4}\)	\(-\frac{ 3}{ 4}-81\lambda\)
\(18\)	\(-9\)	\(\frac{ 11}{ 2}\)	\(-\frac{ 3}{ 2}-162\lambda\)
\(36\)	\(-18\)	\(9\)	\(-2-324\lambda\)

\(-\frac{ 1}{ 6}+I3^{ \frac{ 6}{ 7}}exp(-\frac{ 4}{ 35})\)	\(\frac{ 7}{ 4}+236\lambda\)	\(.38888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888889-.762440866I\)	\(-.287127832\)
\(\frac{ 4}{ 3}\)	\(\frac{ 5}{ 3}\)	\(-\frac{ 4}{ 9}\)	\(.38888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888889+.762440866I\)
\(-2\)	\(3\)	\(\frac{ 5}{ 3}\)	\(\frac{ 7}{ 4}-236\lambda\)
\(-4\)	\(-2\)	\(\frac{ 4}{ 3}\)	\(-\frac{ 1}{ 6}-I3^{ \frac{ 6}{ 7}}exp(-\frac{ 4}{ 35})\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

Explicit solution

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

\(1\)	\(-1\)	\(\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 6}\)
\(0\)	\(1\)	\(-1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(0\)	\(1\)	\(-1\)
\(0\)	\(0\)	\(0\)	\(1\)

\(1+162\lambda\)	\(0\)	\(27\lambda\)	\(.34239575e-1\)
\(3\)	\(1\)	\(\frac{ 1}{ 2}\)	\(-27\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(18\)	\(0\)	\(3\)	\(1-162\lambda\)

\(-162\lambda\)	\(6\)	\(1\)	\(1\)
\(-3\)	\(-9\)	\(-1\)	\(0\)
\(0\)	\(18\)	\(0\)	\(0\)
\(-18\)	\(0\)	\(0\)	\(0\)

\(\frac{ 7}{ 6}-\frac{ 10413121032779260648346948458175935308908666354681396315952835358306843793103279948239733597798534695}{ 4552537122981102476060103483058775865936383743190033321979354230308802848949268282146611842149780696}I\)	\(\frac{ 5}{ 3}\)	\(\frac{ 1}{ 2}\)	\(1\)
\(-\frac{ 4}{ 3}\)	\(0\)	\(-1\)	\(0\)
\(2\)	\(-4\)	\(0\)	\(0\)
\(4\)	\(0\)	\(0\)	\(0\)