Calabi-Yau differential operator database v.3.0

1

New Number: 2.20 | AESZ: 133 | Superseeker: 12 -3284/3 | Hash: 4c9628f7dd48f4e9e6ec75303e557389

Degree: 2

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

Coefficients of the holomorphic solution: 1, 12, 324, 8400, 44100, ...
--> OEIS
Normalized instanton numbers (n₀=1): 12, -42, -3284/3, -20538, -103776, ... ; Common denominator:...

Discriminant

$1-144z+6912z^2$

Local exponents

$0$ $\frac{ 1}{ 96}-\frac{ 1}{ 288}\sqrt{ 3}I$ $\frac{ 1}{ 96}+\frac{ 1}{ 288}\sqrt{ 3}I$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 3}{ 2}$
$0$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 96}-\frac{ 1}{ 288}\sqrt{ 3}I\)	\(\frac{ 1}{ 96}+\frac{ 1}{ 288}\sqrt{ 3}I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product A*f
Explicit solution not yet verified

Integral instantons: ,...
Coefficients of the Yukawa coupling: 1, 12, -324, -29544, -1314756, -12971988, 2033927928, 146587697352,...
Coefficients of the q-coordinate : 0, 1, -36, 1134, -24912, 564597, -11502648, 173126610,...

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

Explicit solution

$A_{n}=\dbinom{2n}{n}^2\sum_{k=0}^{n}(-1)^{k}3^{n-3k}\dbinom{n}{3k}\frac{(3k)!}{k!^3}$

Maple LaTex

No topological data

Monodromy (with respect to Frobenius basis)

$1$	$-1$	$\frac{ 1}{ 2}$	$-\frac{ 1}{ 6}$
$0$	$1$	$-1$	$\frac{ 1}{ 2}$
$0$	$0$	$1$	$-1$
$0$	$0$	$0$	$1$

copy data

$\frac{ 1}{ 6}+120\lambda$	$-\frac{ 5}{ 36}+20\lambda$	$-\frac{ 5}{ 54}+\frac{ 40}{ 3}\lambda$	$-.9896550e-2+\frac{ 50}{ 9}\lambda$
$4+\frac{ 7}{ 1000000000}I$	$\frac{ 5}{ 3}-\frac{ 1}{ 500000000}I$	$\frac{ 4}{ 9}-\frac{ 1}{ 500000000}I$	$\frac{ 5}{ 54}-\frac{ 40}{ 3}\lambda$
$-6.-.11e-7I$	$-1+\frac{ 3}{ 1000000000}I$	$\frac{ 1}{ 3}+\frac{ 3}{ 1000000000}I$	$-\frac{ 5}{ 36}+20\lambda$
$36.000000031+.67e-7I$	$-24I3^{ \frac{ 19}{ 41}}\lambdaPi^3-\frac{ 1}{ 62500000}I$	$4-\frac{ 9}{ 500000000}I$	$\frac{ 11}{ 6}-120\lambda$

copy data

$\frac{ 11}{ 6}+120\lambda$	$-\frac{ 5}{ 36}-20\lambda$	$\frac{ 5}{ 54}+\frac{ 40}{ 3}\lambda$	$-.9896550e-2-\frac{ 50}{ 9}\lambda$
$4$	$\frac{ 1}{ 3}$	$\frac{ 4}{ 9}$	$-\frac{ 5}{ 54}-\frac{ 40}{ 3}\lambda$
$6$	$-1$	$\frac{ 5}{ 3}$	$-\frac{ 5}{ 36}-20\lambda$
$36$	$-6$	$4$	$\frac{ 1}{ 6}-120\lambda$

copy data

Basis of the Doran-Morgan lattice

$\frac{ 5}{ 6}-120\lambda$	$\frac{ 78000000031}{ 6000000000}+\frac{ 71}{ 3000000000}I$	$\frac{ 302400000483599999147}{ 259200000446400001090}+\frac{ 1124400000899}{ 259200000446400001090}I$	$1$
$-4-\frac{ 7}{ 1000000000}I$	$-\frac{ 48000000031}{ 2000000000}-\frac{ 89}{ 2000000000}I$	$-\frac{ 648000001115999998236}{ 648000001116000002725}-\frac{ 2412000002077}{ 648000001116000002725}I$	$0$
$6+\frac{ 11}{ 1000000000}I$	$\frac{ 36000000031}{ 1000000000}+\frac{ 67}{ 1000000000}I$	$0$	$0$
$-\frac{ 36000000031}{ 1000000000}-\frac{ 67}{ 1000000000}I$	$0$	$0$	$0$

copy data

2

New Number: 4.32 | AESZ: 356 | Superseeker: -14 -196 | Hash: e73c971c3ed3a4fd581234510642c285

Degree: 4

$\theta^4-2 x\left(236\theta^4+472\theta^3+577\theta^2+341\theta+78\right)+2^{2} x^{2}\left(20836\theta^4+83344\theta^3+150531\theta^2+134374\theta+48664\right)-2^{6} 3 x^{3}\left(33984\theta^4+203904\theta^3+487828\theta^2+545916\theta+237757\right)+2^{10} 3^{2} x^{4}(12\theta+19)(12\theta+23)(12\theta+25)(12\theta+29)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 156, 21062, 2714208, 342489420, ...
--> OEIS
Normalized instanton numbers (n₀=1): -14, -42, -196, -1218, -208446/5, ... ; Common denominator:...

Discriminant

$(128z-1)^2(108z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 128}$ $\frac{ 1}{ 108}$ $\infty$
$0$ $0$ $0$ $\frac{ 19}{ 12}$
$0$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$ $\frac{ 23}{ 12}$
$0$ $1$ $1$ $\frac{ 25}{ 12}$
$0$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 29}{ 12}$

\(0\)	\(\frac{ 1}{ 128}\)	\(\frac{ 1}{ 108}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 19}{ 12}\)
\(0\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)	\(\frac{ 23}{ 12}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 25}{ 12}\)
\(0\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 29}{ 12}\)

Note:

Sporadic YY-Operator.

3

New Number: 5.46 | AESZ: 243 | Superseeker: -42 -41706 | Hash: 93c30005b5a976a2b7c5206d5e679a45

Degree: 5

$\theta^4+x\left(295\theta^4+572\theta^3+424\theta^2+138\theta+17\right)+2 x^{2}\left(843\theta^4+744\theta^3-473\theta^2-481\theta-101\right)+2 x^{3}\left(1129\theta^4-516\theta^3-725\theta^2-159\theta+4\right)-3 x^{4}\left(173\theta^4+352\theta^3+290\theta^2+114\theta+18\right)-3^{2} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, -17, 1549, -215585, 36505501, ...
--> OEIS
Normalized instanton numbers (n₀=1): -42, 875, -41706, 2954224, -257813864, ... ; Common denominator:...

Discriminant

$-(z^3+57z^2-289z-1)(3z+1)^2$

Local exponents

≈$-61.684843$ $-\frac{ 1}{ 3}$ ≈$-0.003458$ $0$ ≈$4.688301$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $0$ $1$ $1$
$1$ $3$ $1$ $0$ $1$ $1$
$2$ $4$ $2$ $0$ $2$ $1$

≈\(-61.684843\)	\(-\frac{ 1}{ 3}\)	≈\(-0.003458\)	\(0\)	≈\(4.688301\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)
\(1\)	\(3\)	\(1\)	\(0\)	\(1\)	\(1\)
\(2\)	\(4\)	\(2\)	\(0\)	\(2\)	\(1\)

Note:

A-incarnation: $7 \times 7$ linear Pfaffian in $P^7$.
There is a second MUM point at infinity, associated to
the 7 fold linear section of $G(2,7)$ AESZ 27/5.7

\(\frac{ 1}{ 6}+120\lambda\)	\(-\frac{ 5}{ 36}+20\lambda\)	\(-\frac{ 5}{ 54}+\frac{ 40}{ 3}\lambda\)	\(-.9896550e-2+\frac{ 50}{ 9}\lambda\)
\(4+\frac{ 7}{ 1000000000}I\)	\(\frac{ 5}{ 3}-\frac{ 1}{ 500000000}I\)	\(\frac{ 4}{ 9}-\frac{ 1}{ 500000000}I\)	\(\frac{ 5}{ 54}-\frac{ 40}{ 3}\lambda\)
\(-6.-.11e-7I\)	\(-1+\frac{ 3}{ 1000000000}I\)	\(\frac{ 1}{ 3}+\frac{ 3}{ 1000000000}I\)	\(-\frac{ 5}{ 36}+20\lambda\)
\(36.000000031+.67e-7I\)	\(-24I3^{ \frac{ 19}{ 41}}\lambdaPi^3-\frac{ 1}{ 62500000}I\)	\(4-\frac{ 9}{ 500000000}I\)	\(\frac{ 11}{ 6}-120\lambda\)

\(\frac{ 11}{ 6}+120\lambda\)	\(-\frac{ 5}{ 36}-20\lambda\)	\(\frac{ 5}{ 54}+\frac{ 40}{ 3}\lambda\)	\(-.9896550e-2-\frac{ 50}{ 9}\lambda\)
\(4\)	\(\frac{ 1}{ 3}\)	\(\frac{ 4}{ 9}\)	\(-\frac{ 5}{ 54}-\frac{ 40}{ 3}\lambda\)
\(6\)	\(-1\)	\(\frac{ 5}{ 3}\)	\(-\frac{ 5}{ 36}-20\lambda\)
\(36\)	\(-6\)	\(4\)	\(\frac{ 1}{ 6}-120\lambda\)

\(\frac{ 5}{ 6}-120\lambda\)	\(\frac{ 78000000031}{ 6000000000}+\frac{ 71}{ 3000000000}I\)	\(\frac{ 302400000483599999147}{ 259200000446400001090}+\frac{ 1124400000899}{ 259200000446400001090}I\)	\(1\)
\(-4-\frac{ 7}{ 1000000000}I\)	\(-\frac{ 48000000031}{ 2000000000}-\frac{ 89}{ 2000000000}I\)	\(-\frac{ 648000001115999998236}{ 648000001116000002725}-\frac{ 2412000002077}{ 648000001116000002725}I\)	\(0\)
\(6+\frac{ 11}{ 1000000000}I\)	\(\frac{ 36000000031}{ 1000000000}+\frac{ 67}{ 1000000000}I\)	\(0\)	\(0\)
\(-\frac{ 36000000031}{ 1000000000}-\frac{ 67}{ 1000000000}I\)	\(0\)	\(0\)	\(0\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

Explicit solution

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:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice