Stable Evaluation of Lefschetz Thimble Intersection Numbers: Towards Real-Time Path Integrals

TL;DR

The paper tackles the sign problem in multivariable oscillatory integrals by leveraging Picard-Lefschetz theory to decompose the real integration domain into Lefschetz thimbles and compute intersection numbers $n_\sigma$ with the original cycle. It introduces a robust multiple-shooting method to solve upward-flow equations from saddles $z_\sigma$ to the integration cycle, enabling simultaneous determination of flow existence and the sign of $n_\sigma$ even in high dimensions. Demonstrations on a three-variable Airy-type integral and a discretized double-well path integral show excellent agreement with saddle-point estimates and reveal nontrivial contributions from complex saddles, including Stokes phenomena. The approach is scalable, yields practical runtimes on standard hardware, and holds potential for extensions to degenerate saddles, higher precision arithmetic, and broader oscillatory-integral problems in physics and mathematics.

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 $20$ 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.

Figures (10)

• Figure 1: The upward flow from one of the saddle points to the original integration cycle for $\alpha=3.01$. Each panel shows the projection on the $(\Re z_i,\Im z_i)$-plane. The orange circle is the saddle point and the blue curve is the flow we obtained. We also show ${\rm d}z/{\rm d}s$ around the flow with red arrows.
• Figure 2: Comparison between the saddle-point approximation and direct numerical integration for the three-variable Airy-type integral. The colored lines represent contributions from individual saddle points, while the black thick line denotes results from direct numerical integration.
• Figure 3: Upward flow from the saddle point $(n,m) = (4,2)$ to the original integration cycle. The black dashed line denotes the saddle point in the continuum limit, while the orange circles indicate the discretized saddle for $L = 20$. The upward flow trajectory is shown as colored lines, progressing from red ($s = 0$) to purple ($s = s_f$), with points plotted every $25$ steps.
• Figure 4: The same figure as Fig. \ref{['fig:airy_flow']}, but for those used in the analysis of convergence. The top panels show a typical convergent case ($\alpha=1.6$, $\delta r=0.01$). The bottom panels show a special case where two saddle points are close and the Jacobian becomes ill-conditioned ($\alpha=2.6$, $\delta r=0.01$). The left, middle, and right panels show the $z_0$, $z_1$, and $z_2$ components of the flow, respectively.
• Figure 5: Convergence of Newton's method. The left panel shows a typical convergent case ($\alpha=1.6$, $\delta r=0.01$, the saddle shown in the top panels of Fig. \ref{['fig:airy_examples']}). The middle panel shows a special case where two saddle points are close and the Jacobian becomes ill-conditioned ($\alpha=2.6$, $\delta r=0.01$, the saddle shown in the bottom panels of Fig. \ref{['fig:airy_examples']}). The right panel shows a non-convergent case where there is no solution ($\alpha=1.6$, $\delta r=0.01$, another saddle specified in the text).