Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models

Understand

Zero-dimensional $O(n)$-symmetric sigma models are studied by using Picard--Lefschetz integration method in the presence of small symmetry-breaking perturbations.

• Due to approximate symmetry, downward flows turn out to show significant structures: They slowly travel along the set of pseudo classical points, and branch into other directions so as to span middle-dimensional integration cycles.
• We propose an efficient way to find such slow motions for computing Lefschetz thimbles.
• In the limit of symmetry restoration, we figure out that only special combinations of Lefschetz thimbles can survive as convergent integration cycles: Other integrations become divergent due to non-compactness of the complexified group of symmetry.