Stable Evaluation of Lefschetz Thimble Intersection Numbers: Towards Real-Time Path Integrals
arXiv:2510.06334 · doi:10.1016/j.physletb.2026.140198
Abstract
We introduce a robust numerical method for determining intersection numbers of Lefschetz thimbles in multivariable settings. Our approach employs the multiple shooting method to solve the upward flow equations from the saddle points to the original integration cycle, which also enables us to determine the signs of the intersection numbers. The method demonstrates stable and reliable performance, and has been tested for systems with up to variables, which can be further extended by adopting quadruple-precision arithmetic. We determine intersection numbers for several complex saddle points in a discretized path integral, providing new insights into the structure of real-time path integrals. The proposed method is broadly applicable to a wide range of problems involving oscillatory integrals in physics and mathematics.
20 pages, 10 figures, 4 tables; added link to the Mathematica code; Published version
References in corpus (17)
- High density QCD on a Lefschetz thimble?
- Lorentzian Quantum Cosmology
- Monte Carlo simulations on the Lefschetz thimble: taming the sign problem
- Hybrid Monte Carlo on Lefschetz Thimbles -- A study of the residual sign problem
- No smooth beginning for spacetime
- Structure of Lefschetz thimbles in simple fermionic systems
- Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling
- Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density
- Lefschetz thimbles and stochastic quantisation: Complex actions in the complex plane
- Functional methods for false vacuum decay in real time
- On exact-WKB analysis, resurgent structure, and quantization conditions
- Vacuum decay in the Lorentzian path integral
- Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points
- Truly Lorentzian quantum cosmology
- False vacuum decay of excited states in finite-time instanton calculus
- All-order Resurgence from Complexified Path Integral in a Quantum Mechanical System with Integrability
- Acquiring the Lefschetz thimbles: efficient evaluation of the diffraction integral for lensing in wave optics