F4 restricted to hypersurface singularity

References

  1. Horn data
  2. A numerical analysis of $F_4$ on singularity (link to google colaboratory)
  3. 2022-12-23-F4.rr (Program to generate the ODE used above).
  4. Vsevolod Chestnov, Saiei J. Matsubara-Heo, Henrik J. Munch, Nobuki Takayama, Restrictions of Pfaffian Systems for Feynman Integrals, doi:10.1007/JHEP11(2023)202
  5. mt_mm package manual

Data format

A Pfaffian system for Appell's $F_4$ restricted to the hypersurface singularity. The ODE on the hypersurface singularity $f(x,y)=0$ is given by two square matrices
$P_1$ and $P_2$
whose elements lie in the quotient ring $K[x,y]/f$. If we parametrize the singularity $f(x,y)=0$ as
$x=x(t), y=y(t)$
then the ODE is given as
$\frac{dF}{dt}= \left( P_1 x'(t) +P_2 y'(t) \right) F $
where $F$ is a column vector valued function.

Under construction

The following data are given for some special parameters. For other parameters, please use the programs 2024-01-05-F4.rr and 2023-03-29-rest-hs.rr .

  1. Differential operators: $\partial_x (x \partial_x + c_1-1) - (x \partial_x+y \partial_y+a)(x \partial_x+y \partial_y+b)$, $\partial_y (y \partial_y + c_2-1) - (x \partial_x+y \partial_y+a)(x \partial_x+y \partial_y+b)$.
  2. Singularity to restrict: $(x-y)^2-2(x+y)+1$.
  3. Parameter values: $a=2/3$, $b=c_1=c_2=-1/3$.

$P_1$

In Risa/Asir format. 



 $P_2$

In Risa/Asir format.