# 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: `lefschetz.zip`

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We implemented our algorithms in julia (v1.8.3) using the packages HomotopyContinuation.jl (v2.6.4) and Oscar.jl (v0.10.0). We include our implementation in this repository.

To download the code, click here `code.zip`

. The results of the computations reported in Section 6 can be found in txt format here `results_Sec6.zip`

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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)