Calabi-Yau differential operator database v.3.0

1

New Number: 2.8 | AESZ: 63 | Superseeker: 684 195638820 | Hash: 06c1a4c0aa33f5051126908a9898430d

Degree: 2

$\theta^4-2^{2} 3 x(6\theta+1)(6\theta+5)(11\theta^2+11\theta+3)-2^{4} 3^{2} x^{2}(6\theta+1)(6\theta+5)(6\theta+7)(6\theta+11)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 180, 263340, 600359760, 1674535082220, ...
--> OEIS
Normalized instanton numbers (n₀=1): 684, 253314, 195638820, 225040578570, 319342448936304, ... ; Common denominator:...

Discriminant

$1-4752z-186624z^2$

No data for singularities

Note:

Hadamard product D*b

	Gopakumar-Vafa invariants
g=0	3420, 1266570, 978194100, 1125202892850, 1596712244681520, 2593390973297156910, 4623712594996202607060, 8826751501399749729515490,...
g=1	1, -611, -3437965, 524537945, 45789504955348, 237013737984797903, 922140400732269341707, 3194302059028418771226413,...
g=2	,...

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-310\)	\(62\)	\(5\)	\(6\)

2

New Number: 3.14 | AESZ: | Superseeker: 444 19050964 | Hash: dc96aa2da269d989ee90c49dab6a9c5a

Degree: 3

$\theta^4-2^{2} x\left(452\theta^4+920\theta^3+633\theta^2+173\theta+17\right)-2^{4} x^{2}(4\theta+3)(3808\theta^3+10504\theta^2+8884\theta+1635)-2^{8} 11^{2} x^{3}(4\theta+1)(4\theta+3)(4\theta+7)(4\theta+9)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 68, 42220, 38866320, 43812369900, ...
--> OEIS
Normalized instanton numbers (n₀=1): 444, 57104, 19050964, 9432910668, 5781274591408, ... ; Common denominator:...

Discriminant

$-(1936z-1)(1+64z)^2$

Local exponents

$-\frac{ 1}{ 64}$ $0$ $\frac{ 1}{ 1936}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 4}$
$\frac{ 1}{ 2}$ $0$ $1$ $\frac{ 3}{ 4}$
$1$ $0$ $1$ $\frac{ 7}{ 4}$
$\frac{ 3}{ 2}$ $0$ $2$ $\frac{ 9}{ 4}$

\(-\frac{ 1}{ 64}\)	\(0\)	\(\frac{ 1}{ 1936}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(\frac{ 3}{ 4}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 7}{ 4}\)
\(\frac{ 3}{ 2}\)	\(0\)	\(2\)	\(\frac{ 9}{ 4}\)

Note:

This is operator Pi = 3.14 (approx.), equivalent to AESZ 238.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-100\)	\(38\)	\(5\)	\(4\)

3

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

Note:

Explicit solution

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

\(-\frac{ 18778337}{ 12500000}I\)	\(\frac{ 41}{ 12}\)	\(1\)	\(1\)
\(-\frac{ 31}{ 12}\)	\(-\frac{ 5}{ 2}\)	\(-1\)	\(0\)
\(0\)	\(5\)	\(0\)	\(0\)
\(-5\)	\(0\)	\(0\)	\(0\)

\(-100\lambda\)	\(\frac{ 29}{ 12}\)	\(1\)	\(1\)
\(-\frac{ 19}{ 12}\)	\(-\frac{ 5}{ 2}\)	\(-1\)	\(0\)
\(0\)	\(5\)	\(0\)	\(0\)
\(-5\)	\(0\)	\(0\)	\(0\)