Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence

TL;DR

This work argues that a correct semiclassical treatment of Euclidean path integrals requires complexification of the action, measure, and field configurations, with complex saddles described by holomorphic Newton equations and organized by Picard-Lefschetz theory. By studying quantum mechanical models with bosonic and fermionic degrees of freedom and graded Hilbert spaces, the authors derive exact complex and real saddle solutions (bounces and bions) in both double-well and periodic (Sine-Gordon) potentials, and show how complex saddles, including multi-valued actions and monodromies, are essential for obtaining results compatible with SUSY and for capturing nonperturbative physics via resurgence and HTA. They connect these exact saddles to approximate quasi-zero-mode thimbles, demonstrating that thimble integrals reproduce key nonperturbative features and hinting at their relevance for quantum field theories, where complex saddles may underlie approximate bion saddles. The study also develops a graded (multi-flavor) formalism clarifying how fermions modify potentials, lift degeneracies, and yield isospectral sectors, thereby providing a coherent framework for integrating fermionic determinants into semiclassical analyses. Collectively, the results establish complexified path integrals as a robust and necessary tool for understanding ground-state structures, monodromies, and resurgent cancellations in both quantum mechanics and potentially QFT. The work also situates these ideas within the broader literature on analytic continuation and complex saddles, emphasizing the role of HTA and resurgent transseries in achieving physically meaningful, real observables.

Abstract

We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on ${\mathbb R}^3\times S^1$, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-degenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct non-perturbative structure of these models. In the supersymmetric limit the complex solutions govern the ground state properties, and their contribution to the semiclassical expansion is necessary to obtain consistency with the supersymmetry algebra. The multi-valuedness of the action is either related to the hidden topological angle or to the resurgent cancellation of ambiguities. We also show that in the approximate multi-instanton description the integration over the complex quasi-zero mode thimble produces the most salient features of the exact solutions. While exact complex saddles are more difficult to construct in quantum field theory, the relation to the approximate thimble construction suggests that such solutions may be underlying some remarkable features of approximate bion saddles in quantum field theories.

Figures (31)

• Figure 1: Bosonic potential $V_{\rm bos}(x)$ (red dashed) vs. quantum modified potential (black solid) $V(x)$ for double-well system. The quantum modified potential (\ref{['eq:dw-mod']}) is obtained upon quantizing the fermions and projecting to a fermion number (or spin) eigenstate.
• Figure 2: Bosonic potential $V_{\rm bos}(x)$ (red dashed) vs. quantum modified potential (black solid) $V(x)$ for the periodic potential. The quantum modified potential (\ref{['eq:sg-mod']}) is obtained upon quantizing the fermions and projecting to a fermion number (or spin) eigenstate.
• Figure 3: Inverted quantum modified potential in a theory with a symmetric bosonic double-well potential. There are no real finite action saddle configurations contributing to ground state properties, but there are exact complex saddle solutions. The Euclidean description of the vacuum is a dilute gas of these complex saddle points, which we call complex bions. The complex bion is necessary in order to explain the non-perturbatively induced vacuum energy of the corresponding $N_f=1$ SUSY QM model.
• Figure 4: Inverted quantum modified potential in a theory with a periodic bosonic potential. There are exact real and complex saddle solutions in the quantum modified Sine-Gordon model. Both the real and complex bions are necessary in order to explain the non-perturbative vanishing of the vacuum energy of the corresponding $N_f=1$ SUSY QM model. The Euclidean description of the vacuum is a dilute gas of real and complex bions.
• Figure 5: Fermion number vs. spin representation for the multi-flavor theory with Lagrangian \ref{['lag']}. $N_f$ is chosen to be odd. The system describes a particle with internal $({ {\frac{1}{2}}})^{N_f}$-spin.