Twisted Cohomology and Likelihood Ideals
ABSTRACT: A likelihood function on a smooth very affine variety gives rise to a twisted de Rham complex. We show how its top cohomology vector space degenerates to the coordinate ring of the critical points defined by the likelihood ideal. We obtain a basis for cohomology from a basis of this coordinate ring. We investigate the dual picture, where twisted cycles correspond to critical points. We show how to expand a twisted cocycle in terms of a basis, and apply our methods to Feynman integrals from physics.
The following picture illustrates Lefschetz thimbles, which constitute a natural basis for twisted homology on the one dimensional torus with the third roots of unity removed.
This picture was generated using Mathematica. The script can be downloaded here:
Project page created: 31/01/2023
Project contributors: Saiei-Jaeyeong Matsubara-Heo and Simon Telen
Corresponding author of this page: Simon Telen, Simon.Telen@mis.mpg.de
Software used: Julia (Version 1.8.3), Mathematica (Version 12.3.1)