Stable Evaluation of Lefschetz Thimble Intersection Numbers: Towards Real-Time Path Integrals
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917234019926016 |
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| author | Shoji, Yutaro Trailović, Katarina |
| author_facet | Shoji, Yutaro Trailović, Katarina |
| contents | 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 $20$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_06334 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable Evaluation of Lefschetz Thimble Intersection Numbers: Towards Real-Time Path Integrals Shoji, Yutaro Trailović, Katarina High Energy Physics - Theory High Energy Physics - Lattice Mathematical Physics 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 $20$ 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. |
| title | Stable Evaluation of Lefschetz Thimble Intersection Numbers: Towards Real-Time Path Integrals |
| topic | High Energy Physics - Theory High Energy Physics - Lattice Mathematical Physics |
| url | https://arxiv.org/abs/2510.06334 |