Simulating Yang-Mills theories with a complex coupling

TL;DR

This paper introduces a new simulation method for Yang-Mills theories with complex couplings using Lefschetz thimbles, potentially enabling more effective Monte Carlo calculations in models with sign problems.

Contribution

The authors develop a novel Lefschetz thimble-based simulation strategy for complex Yang-Mills theories, adaptable to QCD at finite density and real-time scenarios.

Findings

Algorithm demonstrated on a (1+1)-D U(1) model.

Exponential performance improvement over standard reweighting.

Residual sign problem remains but is mitigated.

Abstract

We propose a novel simulation strategy for Yang-Mills theories with a complex coupling, based on the Lefschetz thimble decomposition. We envisage, that the approach developed in the present work, can also be adapted to QCD at finite density, and real time simulations. Simulations with Lefschetz thimbles offer a potential solution to sign problems in Monte Carlo calculations within many different models with complex actions. We discuss the structure of Generalized Lefschetz thimbles for pure Yang-Mills theories with a complex gauge coupling $\beta$ and show how to incorporate the gauge orbits. We propose to simulate such theories on the union of the tangential manifolds to the relevant Lefschetz thimbles attached to the critical manifolds of the Yang-Mills action. We demonstrate our algorithm on a (1+1)-dimensional U(1) model and discuss how, starting from the main thimble result, successive subleading thimbles can be taken into account via a reweighting approach. While we face a residual sign problem, our novel approach performs exponentially better than the standard reweighting approach.