Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles

TL;DR

This work develops a resurgence-based framework for decoding the semiclassical expansion in the principal chiral model (PCM), a two-dimensional asymptotically free theory with trivial topology (no instantons). By leveraging adiabatic continuity to a small-S^1 circle, the authors reveal an infinite class of non-perturbative saddles—fractons and their composites—that govern mass generation and IR renormalon structure, despite the absence of traditional topological charges. A central result is that perturbative large-order behavior encodes NP data through Stokes phenomena and that neutral bions cancel perturbative ambiguities, realized both analytically in a zero-dimensional prototype and geometrically via Lefschetz thimbles. The emergent topological structure organizes NP saddles into a resurgence triangle, linking fracton interactions to mass gaps and offering a concrete semi-classical pathway to understanding renormalons in PCM and related theories. The work also introduces the concept of Borel flow, describing how singularities in the Borel plane evolve with the circle radius, suggesting a unified picture connecting small-L semiclassical physics to large-L strongly coupled dynamics.

Abstract

Resurgence theory implies that the non-perturbative (NP) and perturbative (P) data in a QFT are quantitatively related, and that detailed information about non-perturbative saddle point field configurations of path integrals can be extracted from perturbation theory. Traditionally, only stable NP saddle points are considered in QFT, and homotopy group considerations are used to classify them. However, in many QFTs the relevant homotopy groups are trivial, and even when they are non-trivial they leave many NP saddle points undetected. Resurgence provides a refined classification of NP-saddles, going beyond conventional topological considerations. To demonstrate some of these ideas, we study the $SU(N)$ principal chiral model (PCM), a two dimensional asymptotically free matrix field theory which has no instantons, because the relevant homotopy group is trivial. Adiabatic continuity is used to reach a weakly coupled regime where NP effects are calculable. We then use resurgence theory to uncover the existence and role of novel `fracton' saddle points, which turn out to be the fractionalized constituents of previously observed unstable `uniton' saddle points. The fractons play a crucial role in the physics of the PCM, and are responsible for the dynamically generated mass gap of the theory. Moreover, we show that the fracton-anti-fracton events are the weak coupling realization of 't Hooft's renormalons, and argue that the renormalon ambiguities are systematically cancelled in the semi-classical expansion. Our results motivate the conjecture that the semi-classical expansion of the path integral can be geometrized as a sum over Lefschetz thimbles.

Figures (13)

• Figure 1: Left: Lefschetz thimbles at $\lambda=e^{i\theta}$ with $\theta=0^-$: ${\cal J}_0 + {\cal J}_1$. Right: At $\theta=0^{+}$. ${\cal J}_0 - {\cal J}_1.$ We take $\theta=\mp 0.1$ to ease visualization.
• Figure 2: The original integration cycle as a linear combination of Lefschetz thimbles at $\theta=0^-$ and $\theta=0^+$. $\theta=0$ is a Stokes line.
• Figure 3: Stokes phenomenon (wall-crossing) at $\theta=0$: ${\cal J}_0 \rightarrow {\cal J}_0 - 2 {\cal J}_1$, while ${\cal J}_1 \rightarrow {\cal J}_1$. There is also a Stokes line at $\theta=\pi$ where ${\cal J}_1$ jumps and ${\cal J}_0$ does not.
• Figure 4: The right Borel resummation can be rewritten as the sum of the left Borel resummation plus the contribution coming from the Hankel contour $\gamma$, coming from $t\to-\infty$, circling around the branch cut starting at $t=1/2$ and going back to $+\infty$.
• Figure 5: Top: With a thermal compactification on ${\mathbb R}\times S^1_\beta$, there is a rapid-crossover from an $F/N^2 \sim O(1)$ (deconfined) behavior of the free energy to $F/N^2 = 0$ confined regime, which becomes a genuine phase transition at $N=\infty$. Even at finite-$N$, however, the quantitative behavior of the theory changes dramatically between these two regimes, despite the fact that there is no sharp phase transition in a finite volume. Bottom: By using spatial compactification, we find a unique small-$L$ limit in which "free energy" scales as $F/N^2 = 0$. The behavior of the theory does not change dramatically from small-$L$ to large-$L$. This is the idea of adiabatic continuity. Since the small-$L$ theory is weakly coupled thanks to asymptotic freedom, it is NP-calculable, and the knowledge gained therein is continuously connected to the physics of decompactified theory on ${\mathbb R}^2$.