Fast Estimator of jacobians in Monte Carlo Integration on Lefschetz Thimbles

Understand

A solution to the sign problem is the so-called "Lefschetz thimble approach" where the domain of integration for field variables in the path integral is deformed from the real axis to a sub-manifold in the complex space.

• For properly chosen sub-manifolds ("thimbles") the sign problem disappears or is drastically alleviated.
• The parametrization of the thimble by real coordinates require the calculation of a jacobian with a computational cost of order O(V^3), where V is proportional to the spacetime volume.
• In this note we propose two estimators for this jacobian with a computational cost of order O(V).