Calabi-Yau differential operator database v.3.0

1

New Number: 4.36 | AESZ: 109 | Superseeker: 1434/7 18676572/7 | Hash: bca2938ac7fa09f5bdc395cab75caf82

Degree: 4

$7^{2} \theta^4-2 3 7 x\left(1272\theta^4+2508\theta^3+1779\theta^2+525\theta+56\right)+2^{2} 3 x^{2}\left(43704\theta^4+38088\theta^3-25757\theta^2-20608\theta-3360\right)-2^{4} 3^{3} x^{3}\left(2736\theta^4-1512\theta^3-1672\theta^2-357\theta-14\right)-2^{6} 3^{5} x^{4}(3\theta+1)(2\theta+1)^2(3\theta+2)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 48, 15840, 8148000, 5126536800, ...
--> OEIS
Normalized instanton numbers (n₀=1): 1434/7, 14718, 18676572/7, 4988009280/7, 1646787631350/7, ... ; Common denominator:...

Discriminant

$-(432z^2+1080z-1)(-7+36z)^2$

Local exponents

$-\frac{ 5}{ 4}-\frac{ 13}{ 18}\sqrt{ 3}$ $0$ $s_1$ $s_2$ $-\frac{ 5}{ 4}+\frac{ 13}{ 18}\sqrt{ 3}$ $\frac{ 7}{ 36}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 3}$
$1$ $0$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $3$ $\frac{ 1}{ 2}$
$2$ $0$ $2$ $2$ $2$ $4$ $\frac{ 2}{ 3}$

\(-\frac{ 5}{ 4}-\frac{ 13}{ 18}\sqrt{ 3}\)	\(0\)	\(s_1\)	\(s_2\)	\(-\frac{ 5}{ 4}+\frac{ 13}{ 18}\sqrt{ 3}\)	\(\frac{ 7}{ 36}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(3\)	\(\frac{ 1}{ 2}\)
\(2\)	\(0\)	\(2\)	\(2\)	\(2\)	\(4\)	\(\frac{ 2}{ 3}\)

Note:

Sporadic Operator.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-120\)	\(46\)	\(7\)	\(5\)

2

New Number: 4.46 | AESZ: 237 | Superseeker: 208 1218192 | Hash: 52c18dd4477f6548dd3b185e97b94c20

Degree: 4

$\theta^4-2^{4} x\left(46\theta^4+128\theta^3+91\theta^2+27\theta+3\right)-2^{9} 3 x^{2}\left(74\theta^4-16\theta^3-231\theta^2-127\theta-20\right)+2^{14} 3^{2} x^{3}\left(14\theta^4+216\theta^3+175\theta^2+51\theta+5\right)+2^{19} 3^{3} x^{4}(3\theta+1)(2\theta+1)^2(3\theta+2)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 48, 12240, 4972800, 2489533200, ...
--> OEIS
Normalized instanton numbers (n₀=1): 208, 5874, 1218192, 220754467, 56417503216, ... ; Common denominator:...

Discriminant

$(864z-1)(64z-1)(1+96z)^2$

Local exponents

$-\frac{ 1}{ 96}$ $0$ $\frac{ 1}{ 864}$ $\frac{ 1}{ 64}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 1}{ 3}$
$1$ $0$ $1$ $1$ $\frac{ 1}{ 2}$
$3$ $0$ $1$ $1$ $\frac{ 1}{ 2}$
$4$ $0$ $2$ $2$ $\frac{ 2}{ 3}$

\(-\frac{ 1}{ 96}\)	\(0\)	\(\frac{ 1}{ 864}\)	\(\frac{ 1}{ 64}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(\frac{ 2}{ 3}\)

Note:

This is operator "4.46" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-78\)	\(44\)	\(8\)	\(5\)

3

New Number: 5.37 | AESZ: 221 | Superseeker: 492/5 872164/5 | Hash: b7ce7a734c057660ce3d6341a7572078

Degree: 5

$5^{2} \theta^4-2^{2} 5 x\left(404\theta^4+1096\theta^3+773\theta^2+225\theta+25\right)-2^{4} x^{2}\left(66896\theta^4+137408\theta^3+101096\theta^2+52800\theta+11625\right)-2^{8} 3 5 x^{3}(2\theta+1)(5672\theta^3+9500\theta^2+8422\theta+2689)-2^{15} 3^{2} x^{4}(2\theta+1)(1208\theta^3+2892\theta^2+2842\theta+969)-2^{20} 3^{3} x^{5}(2\theta+1)(6\theta+5)(6\theta+7)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 20, 2988, 618320, 156299500, ...
--> OEIS
Normalized instanton numbers (n₀=1): 492/5, 10376/5, 872164/5, 91316176/5, 12181916784/5, ... ; Common denominator:...

Discriminant

$-(-1+432z)(16z+1)^2(192z+5)^2$

Local exponents

$-\frac{ 1}{ 16}$ $-\frac{ 5}{ 192}$ $0$ $\frac{ 1}{ 432}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$\frac{ 1}{ 4}$ $1$ $0$ $1$ $\frac{ 5}{ 6}$
$\frac{ 3}{ 4}$ $3$ $0$ $1$ $\frac{ 7}{ 6}$
$1$ $4$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 16}\)	\(-\frac{ 5}{ 192}\)	\(0\)	\(\frac{ 1}{ 432}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 4}\)	\(1\)	\(0\)	\(1\)	\(\frac{ 5}{ 6}\)
\(\frac{ 3}{ 4}\)	\(3\)	\(0\)	\(1\)	\(\frac{ 7}{ 6}\)
\(1\)	\(4\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.37" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-32\)	\(40\)	\(10\)	\(5\)

4

New Number: 1.11 | AESZ: 11 | Superseeker: 324 10792428 | Hash: 8ac8b98b80383c9f0ea125ccd6e6a55d

Degree: 1

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 72, 37800, 31046400, 31216185000, ...
--> OEIS
Normalized instanton numbers (n₀=1): 324, 37260, 10792428, 4580482284, 2405245303584, ... ; Common denominator:...

Discriminant

$1-1728z$

Local exponents

$0$ $\frac{ 1}{ 1728}$ $\infty$
$0$ $0$ $\frac{ 1}{ 4}$
$0$ $1$ $\frac{ 1}{ 3}$
$0$ $1$ $\frac{ 2}{ 3}$
$0$ $2$ $\frac{ 3}{ 4}$

\(0\)	\(\frac{ 1}{ 1728}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(0\)	\(1\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(2\)	\(\frac{ 3}{ 4}\)

Note:

A-incarnation: X(4,6) in P^5(1,1,1,2,2,3)

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-156\)	\(48\)	\(6\)	\(5\)

5

New Number: 1.14 | AESZ: 14 | Superseeker: 1248 683015008 | Hash: 03af56f4ae0cea2c4b219620b08dc49b

Degree: 1

$\theta^4-2^{4} 3 x(6\theta+1)(2\theta+1)^2(6\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 240, 498960, 1633632000, 6558930378000, ...
--> OEIS
Normalized instanton numbers (n₀=1): 1248, 597192, 683015008, 1149904141056, 2394928461766560, ... ; Common denominator:...

Discriminant

$1-6912z$

Local exponents

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

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

Note:

A-incarnation: X(2,6) in P^5(1,1,1,1,1,3)

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-256\)	\(52\)	\(4\)	\(5\)

6

New Number: 1.1 | AESZ: 1 | Superseeker: 575 63441275 | Hash: c86f1c284d8c5119801c6ba1343172bb

Degree: 1

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 120, 113400, 168168000, 305540235000, ...
--> OEIS
Normalized instanton numbers (n₀=1): 575, 121850, 63441275, 48493506000, 45861177777525, ... ; Common denominator:...

Discriminant

$1-3125z$

Local exponents

$0$ $\frac{ 1}{ 3125}$ $\infty$
$0$ $0$ $\frac{ 1}{ 5}$
$0$ $1$ $\frac{ 2}{ 5}$
$0$ $1$ $\frac{ 3}{ 5}$
$0$ $2$ $\frac{ 4}{ 5}$

\(0\)	\(\frac{ 1}{ 3125}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 5}\)
\(0\)	\(1\)	\(\frac{ 2}{ 5}\)
\(0\)	\(1\)	\(\frac{ 3}{ 5}\)
\(0\)	\(2\)	\(\frac{ 4}{ 5}\)

Note:

A-incarnation: $X(5) \subset P^4$
B-incarnation: mirror quintic.
P. Candelas, X. de la Ossa, D. Green, L. Parkes,{\em An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds}, Phys. Lett. B 258 (1991), no.1-2, 118-126.

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

Characteristic classes:

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

Discriminant

Local exponents

Note:

Characteristic classes:

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

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Explicit solution

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice