Kahler: An Implementation of Discrete Exterior Calculus on Hermitian Manifolds

TL;DR

Kahler is a Python library that implements discrete exterior calculus on Hermitian manifolds, enabling complex geometric computations with support for arbitrary dimensions, topologies, and metrics, and demonstrating its applications in physics and engineering.

Contribution

Kahler introduces a highly general framework for discrete exterior calculus on Hermitian manifolds, with automatic parallelization and efficient Cython implementation, expanding computational capabilities in geometric analysis.

Findings

Successfully applied to normal modes of vibrating membranes

Demonstrated electromagnetic resonance computations

Validated convergence on random meshes

Abstract

This paper details the techniques and algorithms implemented in Kahler, a Python library that implements discrete exterior calculus on arbitrary Hermitian manifolds. Borrowing techniques and ideas first implemented in PyDEC, Kahler provides a uniquely general framework for computation using discrete exterior calculus. Manifolds can have arbitrary dimension, topology, bilinear Hermitian metrics, and embedding dimension. Kahler comes equipped with tools for generating triangular meshes in arbitrary dimensions with arbitrary topology. Kahler can also generate discrete sharp operators and implement de Rham maps. Computationally intensive tasks are automatically parallelized over the number of cores detected. The program itself is written in Cython--a superset of the Python language that is translated to C and compiled for extra speed. Kahler is applied to several example problems: normal modes of a vibrating membrane, electromagnetic resonance in a cavity, the quantum harmonic oscillator, and the Dirac-Kahler equation. Convergence is demonstrated on random meshes.