Extend complex Langevin to non-holomorphic actions via holomorphic integrands on alternative complex manifolds

Ascertain whether complex Langevin methods can be correctly applied to systems with originally non-holomorphic actions by defining a holomorphic integrand on a different complex manifold (e.g., a multi-sheeted Riemann surface) that agrees with the real-axis restriction of the action, including verifying correctness criteria and convergence when the action is extended in this manner.

Background

The paper proposes handling non-holomorphic actions by analytically continuing their real-axis restriction to a holomorphic function defined on a suitable multi-sheeted complex manifold, enabling contour deformations protected by Cauchy’s theorem. Complex Langevin methods also rely on holomorphicity assumptions for validity. The authors suggest that a similar manifold-based holomorphic extension might enable complex Langevin to treat non-holomorphic actions, but they do not resolve this question.

This open problem focuses on determining whether the manifold-based holomorphic extension strategy can be leveraged to make complex Langevin valid for actions originally defined with nonanalytic features (e.g., absolute-value potentials), and what theoretical guarantees or conditions would be necessary.