Intersection Numbers of Lefschetz Thimbles

• Intersection numbers of Lefschetz thimbles are topological invariants that encode the sign and multiplicity of how critical points contribute to complex oscillatory integrals.
• The methodology employs a multiple shooting approach with tangent-space propagation to ensure numerical stability and precise determination in high-dimensional systems.
• This framework underpins applications in quantum field theory, statistical mechanics, and complex analysis, enabling accurate decomposition of path integrals into contributing saddle sectors.

Intersection numbers of Lefschetz thimbles are central invariants in the Picard–Lefschetz theory, encoding the topological relationship between the original real integration cycle and the complexified critical point structure of an oscillatory integral. In the context of quantum field theory, statistical mechanics, and complex analysis, they identify which critical points (and their associated “thimbles,” i.e., steepest descent manifolds) actually contribute to the decomposition of a path integral or oscillatory integral, and with what orientation and multiplicity. Formally, for an integral of the form $\int_\mathcal{Y} e^{\mathcal{I}(z)/\hbar} d^L z$, the decomposition is

$$\int_\mathcal{Y} e^{\mathcal{I}(z)/\hbar} d^L z = \sum_{\sigma} n_\sigma \int_{\mathcal{J}_\sigma} e^{\mathcal{I}(z)/\hbar} d^L z,$$

where $\mathcal{J}_\sigma$ is the Lefschetz thimble attached to the saddle point $z_\sigma$, and the integer $n_\sigma$ is the intersection number measuring the homological pairing between the original integration contour and the unstable manifold (anti-thimble) emanating from $z_\sigma$. These coefficients determine the sign and multiplicity with which each saddle's thimble enters the original contour's decomposition and are essential for reconstructing the correct values of highly oscillatory or sign-problematic integrals.

1. Theoretical Framework of Intersection Numbers

Intersection numbers $n_\sigma$ arise from the relative (co)homology theory of complex Morse functions. Consider a holomorphic function $S(z)$ and critical points $z_\sigma$ given by $\partial_z S(z_\sigma) = 0$. The Lefschetz thimble $\mathcal{J}_\sigma$ is the union of all downward (steepest descent) flows for which $\lim_{t\to-\infty} z(t) = z_\sigma$; its dual, the anti-thimble $\mathcal{K}_\sigma$, is defined by the upward (steepest ascent) flow from $z_\sigma$. The decomposition

$$[\mathcal{Y}] = \sum_\sigma n_\sigma [\mathcal{J}_\sigma]$$

pairs the original integration cycle $\mathcal{Y}$ with the set of $\mathcal{J}_\sigma$. The intersection number $n_\sigma$ can be calculated as the signed count of intersections between the anti-thimble $\mathcal{K}_\sigma$ and $\mathcal{Y}$,

$$n_\sigma = \langle \mathcal{Y}, \mathcal{K}_\sigma\rangle.$$

These invariants determine which thimbles contribute and with what orientation. Importantly, their values can change as parameters of $S$ are varied, through Stokes phenomena.

2. Robust Numerical Evaluation: Multiple Shooting Method

Direct integration of the upward (unstable) flow equations to determine $n_\sigma$ is highly sensitive to initial conditions, especially in high dimensions, due to the exponential instability of the flow. The multiple shooting method divides the entire flow interval $[0,s_f]$ into $N$ subintervals and solves the upward flow equation

$$\frac{dz_i}{ds} = \frac{\overline{\partial \mathcal{I}/\partial z_i}}{|\partial \mathcal{I}/\partial z|},$$

on each subinterval independently, reconstructing the full solution by enforcing continuity (matching) conditions at the interval boundaries. The flow is initialized near the saddle $z_\sigma$ with correct tangent space orientation, and the solution is advanced subinterval by subinterval, ensuring that uncertainties do not exponentially amplify over the full trajectory.

A complete set of tangent vectors to the (unstable) manifold is propagated alongside the main flow. At the final time $s_f$, when (and if) intersection with the original real cycle $\mathcal{Y}$ is reached, the sign of the intersection number is given by comparing the orientation of the propagated tangent space to a standard orientation of $\mathcal{Y}$. Explicitly, if $W^+$ are the expanding Takagi vectors at $z_\sigma$, and $J_{Z}^{(N-1)}$ is the Jacobian of the flow, the sign is

$$\text{sign }\det \bigl[ (0\ \mathbb{1})\ J_{Z}^{(N-1)} W^+ \bigr].$$

The full procedure robustly determines $n_\sigma$ and its sign, avoiding catastrophic numerical divergence and working stably up to at least $20$ variables in explicit tests (Shoji et al., 7 Oct 2025). Quadruple-precision arithmetic can be employed if additional numerical stability is required.

3. Mathematical Structure and Boundary Conditions

The initial condition for the upward flow is set by picking a point at a fixed distance $\delta r$ from $z_\sigma$ in the direction of the unstable manifold, satisfying (a) the anchor condition $|Z(0)-Z_\sigma|-\delta r=0$, (b) projection orthogonal to the stable Takagi vectors $(W^-)$ $(W^-)^\top (Z(0)-Z_\sigma) = 0$, and (c) at the flow endpoint $s = s_f$, reality constraints selecting only the real subspace (e.g., for real path integrals, $(0_{L\times L}\ \mathbb{1}_{L\times L}) Z(s_f) = 0$). The stitching variables (the set of intermediate points $Z^{(k)}$ between subintervals) enforce a global solution via Newton's method.

The upward flow distributes seeds on a small $(n_\mathrm{unstable})$-dimensional sphere in the tangent space of the anti-thimble, scanning for intersections with the real cycle. Numerically, the main challenge is solving the corresponding system of equations---the multiple shooting decomposition ensures that error propagation and stiffness are locally controlled, allowing for high precision.

4. Applications: Real-Time Path Integrals and Oscillatory Integrals

Robust determination of intersection numbers is essential in quantum mechanics, quantum field theory, and gravitational lensing, especially when expressing real-time path integrals or other highly oscillatory objects in terms of sum-over-thimbles. In the discretized path integral of a double-well system (with up to $L=20$ time steps), the method successfully tracks all relevant intersection numbers, including their signs (Shoji et al., 7 Oct 2025). In such settings, identification of the correct set of contributing saddles is crucial, as both the value and the phase structure of observables can depend on interference among multiple thimble contributions.

The approach is general and can be extended to Feynman integrals, statistical mechanics with sign problems, or any oscillatory multidimensional integral where the geometry of critical points and thimbles is nontrivial. By stabilizing the upward flow, the method can yield new insight into the structure of quantum tunneling and interference in complexified configuration spaces.

5. Orientation, Topology, and Homological Interpretation

Intersection numbers are homological invariants: they can be computed unambiguously from the relative homology class of $\mathcal{Y}$ with respect to the boundaries at infinity of $\mathbb{C}^{L}$, and the thimbles $\mathcal{J}_\sigma$ form a basis for the relative homology group $H_L(\mathbb{C}^L,L_\infty)$. As such, the intersection matrix $$\langle \mathcal{J}_\sigma,\mathcal{K}_\tau\rangle = \delta_{\sigma\tau}$$ encodes the mutual orthogonality of thimbles and anti-thimbles, and so the procedure is guaranteed to select the correct contributing cycles. The sign (orientation) is fixed by the convention for the ordering of Takagi vectors at each critical point.

As evidenced in (Shoji et al., 7 Oct 2025), the multiple shooting method respects this topological structure, reproducing the theoretically expected intersection numbers even as the dimension and complexity of the problem increases.

6. Broader Implications and Scalability

The method has demonstrated stable locality and accuracy in examples up to $L=20$, with runtimes of $\sim$500 ms for $L=3$ and $\sim$15 s for $L=20$ on a single CPU. The framework is modular, extendable to larger or stiffer problems via precision upgrades, and provides a platform for investigating previously intractable real-time or sign-problematic integrals. Its utility is not restricted to quantum mechanics: highly oscillatory integrals in QCD at finite density, nonequilibrium statistical mechanics, and even cosmological or gravitational lensing problems may benefit.

The ability to determine not only the set of relevant thimbles but also the relative signs and multiplicities of their contributions (i.e., the full vector of intersection numbers) enables controlled analytic continuation, semiclassical analysis, and numerical evaluation of observables where the Sign Problem or Stokes transitions change contributing saddles dynamically.

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Summary Table: Key Elements in Multiple Shooting Intersection Number Computation (Shoji et al., 7 Oct 2025)

Component	Practical Role	Mathematical Tool
Upward flow equation	Tracks dual cycle (anti-thimble) integration	Normalized gradient flow
Multiple shooting	Ensures local numerical stability	Piecewise flow + Newton
Tangent space propagation	Determines orientation/sign	Jacobian determinant
Sign detection	Assigns physical orientation (±1)	Comparison to standard
Application domain	Oscillatory & real-time path integrals	High-dimensional support

7. Conclusion

Intersection numbers of Lefschetz thimbles are decisive in decomposing high-dimensional oscillatory path integrals into their contributing saddle-point sectors. The multiple shooting method combines local numerical stability with tangent-space orientation transport, yielding a robust, high-precision, and general method for computing both the presence and the sign of thimble contributions in complex physical and mathematical systems. This directly advances the applicability and theoretical understanding of thimble-based approaches across quantum theory and beyond (Shoji et al., 7 Oct 2025).