O3O10

Computer-assisted proof of the existence of the O(3) x O(10) non-round Einstein metric, accompanying the paper found here. Arbitrary precision operations are conducted using the FLINT package. To run this code, ensure you are have the latest version of Python installed. You can install required packages using the command

pip install -r requirements.txt

Jupyter notebooks are used to interface with the implementation. These are as follows:

• results.ipynb: Primary Jupyter notebook to generate high-precision solutions and compute all relevant quantities in the associated manuscript.
• proof.ipynb: A Jupyter notebook to verify computed quantities in the proof.
• plots.ipynb: A Jupyter notebook to plot $(A(\alpha),\Omega(\omega))$ for a variety of different $d_1$ and $d_2$.

The implementation itself is contained in four Python scripts:

• cohom1.py: The implementation of the high-precision ODE solver for $\eta$ and $\zeta$ (contained in the class Eta) and $A$ and $\Omega$ (contained in the class AlphaOmega). This file also contains a number of helper functions to generate the required solutions in the manuscript.
• linearized.py: Implementation of the differential of $\eta$ and $\zeta$ with respect to $\alpha$ using finite differences (contained in the class DiffEta). Contains a few helper functions as well.
• arb_cheby.py: Arbitrary-precision implementations of the Chebyshev polynomials with appropriate in-basis manipulations.
• arb_roots.py: Root-finding procedures in arbitrary precision.

Figures are contained in the figs folder. MATLAB scripts to plot $\eta$ and $\zeta$ are contained in the matlab folder.

All precomputed quantities are to be stored in the data folder. You do not need to compute these quantities if you download the precomputed data files in the Releases tab.