Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles (1403.1277v2)

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 unstableuniton' 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.

• The paper applies resurgence theory to decode non-perturbative physics from divergent perturbative QFT expansions, revealing insights beyond traditional instantons.
• Methodologies involve using Lefschetz thimbles for path integrals and identifying novel non-perturbative saddle points, including fractons and unitons.
• The theoretical implications suggest resurgence could offer a universal framework for decoding QFTs, leading to novel computational techniques and insights into mass gaps.

Decoding Perturbation Theory Using Resurgence: Stokes Phenomena, New Saddle Points, and Lefschetz Thimbles

The paper presents a novel approach to understanding quantum field theories (QFTs) through the lens of resurgence theory, detailing a method to decode perturbative expansions in order to gain insights into the non-perturbative aspects of these theories. This paper introduces the framework by examining the principal chiral model (PCM), focusing on the intriguing absence of conventional instantons due to its trivial homotopy group, yet revealing the presence of significant non-perturbative structures.

Key Concepts and Methodologies

1. Resurgence Theory: This mathematical framework connects non-perturbative aspects of a QFT with its perturbative expansions. It posits that perturbative series, despite being divergent, encode all necessary information about non-perturbative phenomena when interpreted in a resurgent framework. The paper shows how resurgence can be applied to analyze the PCM, a model where traditional techniques falter due to its trivial $\pi_2$.
2. Fractons and Unitons: These are newly identified non-perturbative saddle points. The PCM, lacking instantons, harbors "fractons," fractional components of larger structures known as "unitons." Unitons were known in mathematics but lacked a clear quantum interpretation. This research reveals unitons in the PCM are unstable and can fragment into fundamental constituents called fractons.
3. Lefschetz Thimbles: These are used to clarify the analytic continuation and steepest descent of path integrals. For the PCM, thimbles provide the correct basis for redefining integration contours in a complexified field space. This geometrical insight aids in understanding cancellations of non-perturbative ambiguities.
4. Stokes Phenomena: The paper explores how Stokes phenomena are manifested in QFT, specifically how they can lead to (and are resolved by) the introduction of non-perturbative sectors. This is crucial in resolving ambiguities encountered in perturbative expansions.
5. Borel-Ecalle Summation: The methodology extends beyond Borel summation to Borel-Ecalle summation, accounting for non-perturbative completions necessary to retrieve physical observables from divergent series.

Numerical Results and Implications

The analysis predicts novel NP saddle points and configurations like fracton-anti-fracton pairings in the effective small circumference limit of the PCM on $\mathbb{R} \times S^1$. These configurations are crucial for understanding mass gaps and other asymptotic phenomena in QFT, providing a uniform non-perturbative semiclassical definition. Such insights can potentially clarify renormalon contributions and their cancellations with NP saddle amplitudes, deepening our understanding of the structure of $\mathbb{CP}^{N-1}$ models and possibly even Yang-Mills theories.

Theoretical and Practical Implications

The theoretical implications of this work are profound, suggesting that resurgence might offer a universal framework to systematically decode QFTs through perturbative data. Practically, the resurgence approach suggests novel computational techniques that can improve the accuracy and predictability of QFT calculations by incorporating non-perturbative effects.

Future Directions

The implications of these findings suggest multiple future research avenues, including:

• Extending these ideas to other QFTs where topology does not dictate the non-perturbative landscape.
• Exploring the interaction of these methodologies with four-dimensional gauge theories and their associated confinement phenomena.
• Further refining numerical techniques to apply resurgence theory in more complex settings, like finite temperature field theories or models with supersymmetry.

In summary, the paper presents a significant advancement in understanding the intricate web connecting perturbative and non-perturbative regimes in QFTs, potentially revolutionizing approaches in theoretical physics and computational methods.