[Paper Review] Resurgence and Dynamics of O(N) and Grassmannian Sigma Models

Gerald V. Dunne, Mithat Ünsal|arXiv (Cornell University)|May 28, 2015

Black Holes and Theoretical Physics82 references4 citations

TL;DR

This paper applies resurgence theory to O(N) and Grassmannian sigma models on ℝ×S¹, revealing that non-instanton 2d-saddles fractionalize into new saddles tied to the affine root system of 𝔬(N). These fractionalized saddles—particularly neutral bions—cancel perturbative ambiguities and generate the mass gap, providing a non-perturbative continuum definition via resurgent trans-series, with key results confirmed in the large-N limit and connections to renormalons and gauge-string duality.

ABSTRACT

We study the non-perturbative dynamics of the two dimensional ${O(N)}$ and Grassmannian sigma models by using compactification with twisted boundary conditions on $\mathbb R imes S^1$, semi-classical techniques and resurgence. While the $O(N)$ model has no instantons for $N>3$, it has (non-instanton) saddles on $\mathbb R^2$, which we call 2d-saddles. On $\mathbb R imes S^1$, the resurgent relation between perturbation theory and non-perturbative physics is encoded in new saddles, which are associated with the affine root system of the ${\frak o}(N) $ algebra. These events may be viewed as fractionalizations of the 2d-saddles. The first beta function coefficient, given by the dual Coxeter number, can then be intepreted as the sum of the multiplicities (dual Kac labels) of these fractionalized objects. Surprisingly, the new saddles in $O(N)$ models in compactified space are in one-to-one correspondence with monopole-instanton saddles in $SO(N)$ gauge theory on $\mathbb R^3 imes S^1$. The Grassmannian sigma models ${ m Gr}(N, M)$ have 2d instantons, which fractionalize into $N$ kink-instantons. The small circle dynamics of both sigma models can be described as a dilute gas of the one-events and two-events, bions. One-events are the leading source of a variety of non-perturbative effects, and produce the strong scale of the 2d theory in the compactified theory. We show that in both types of sigma models the neutral bion emulates the role of IR-renormalons. We also study the topological theta angle dependence in both the $O(3)$ model and ${ m Gr}(N, M)$, and describe the multi-branched structure of the observables in terms of the theta-angle dependence of the saddle amplitudes.

Motivation & Objective

• To resolve the apparent absence of non-perturbative effects in O(N) models for N≥4, which lack instantons, by identifying alternative finite-action classical solutions.
• To establish a non-perturbative continuum definition of 2d O(N) and Grassmannian sigma models using resurgence and trans-series expansions.
• To clarify the role of neutral bions and kink-instantons in generating the mass gap and canceling perturbative ambiguities, particularly in relation to renormalons.
• To extend the semi-classical analysis of 2d sigma models to include topological theta-angle dependence and multi-branched observables.
• To connect the dynamics of these models to 4d gauge theories via the correspondence between O(N) saddles and SO(N) monopole-instantons on ℝ³×S¹.

Proposed method

• Compactification of the 2d O(N) and Grassmannian sigma models on ℝ×S¹ with twisted boundary conditions to access non-perturbative dynamics.
• Application of resurgence theory to relate perturbative and non-perturbative sectors via trans-series expansions.
• Identification of new non-instanton saddles—fractionalized 2d-saddles—associated with the affine root system of the 𝔬(N) algebra.
• Computation of saddle amplitudes and their contributions to the vacuum energy and mass gap, including neutral and charged bions.
• Use of adiabatic continuity and semi-classical techniques to analyze the structure of the Borel plane and resurgent relations.
• Comparison of results with large-N limits and known results in 4d gauge theories and N=4 SYM, particularly regarding cusp anomalous dimension and condensate cancellation.

Experimental results

Research questions

• RQ1 How can non-perturbative dynamics be consistently described in O(N) models for N≥4, which lack instantons?
• RQ2 What is the origin of the first beta function coefficient β₀=N/2 in O(N) models, and how is it related to the multiplicities of fractionalized saddles?
• RQ3 How do neutral bions in these models emulate the role of infrared renormalons and cancel perturbative ambiguities?
• RQ4 What is the physical interpretation of the multi-branched structure of observables in terms of theta-angle-dependent saddle amplitudes?
• RQ5 How do the new saddles in O(N) models on ℝ×S¹ relate to monopole-instantons in SO(N) gauge theory on ℝ³×S¹?

Key findings

• The O(N) model for N≥4 possesses finite-action 2d-saddles that fractionalize into new saddles associated with the affine root system of 𝔬(N), explaining the non-perturbative structure despite the absence of instantons.
• The first beta function coefficient β₀=N/2 is interpreted as the sum of dual Kac labels (multiplicities) of these fractionalized saddles, providing a geometric origin for the coefficient.
• Neutral bions [B_ii] in the O(N) model are two-fold ambiguous at φ=0± and cancel the perturbative ambiguity in the vacuum energy, ensuring a well-defined non-perturbative result.
• In the O(6) model, the neutral bion amplitude is responsible for canceling the imaginary part of the vacuum energy, consistent with resurgence and the absence of a physical ambiguity.
• The small-circle dynamics of both O(N) and Grassmannian sigma models are described as a dilute gas of one-events (kink-instantons) and two-events (bions), with one-events generating the strong scale of the 2d theory.
• The correspondence between O(N) model saddles and monopole-instantons in SO(N) gauge theory on ℝ³×S¹ provides a direct link between 2d sigma models and 4d gauge theories.