Thimble regularisation of YM fields: crunching a hard problem

TL;DR

This paper explores the challenges of applying thimble regularisation to Yang Mills theories, focusing on 2D models and topological terms, highlighting current progress and unresolved issues.

Contribution

It discusses the potential of 2D YM theories as a testing ground and examines the difficulties in implementing thimble regularisation and topological terms in lattice YM theories.

Findings

Analytic solutions for 2D YM can be expressed as sums over critical points.

Inclusion of topological terms introduces a sign problem in lattice YM.

Current work is ongoing with no definitive solutions yet.

Abstract

Thimble regularisation of Yang Mills theories is still to a very large extent terra incognita. We discuss a couple of topics related to this big issue. 2d YM theories are in principle good candidates as a working ground. An analytic solution is known, for which one can switch from a solution in terms of a sum over characters to a form which is a sum over critical points. We would be interested in an explicit realisation of this mechanism in the lattice regularisation, which is actually quite hard to work out. A second topic is the inclusion of a topological term in the lattice theory, which is the prototype of a genuine sign problem for pure YM fields. For both these challenging problems we do not have final answers. We present the current status of our study.