Resurgence and Dynamics of O(N) and Grassmannian Sigma Models

TL;DR

This work applies resurgence and adiabatic continuity to two-dimensional ${O(N)}$ and Grassmannian sigma models by compactifying on ${\mathbb R}\times S^1$ with twisted boundary conditions, revealing a rich semi-classical structure without relying on conventional instantons for $N\ge 4$. It shows that fractional kink-instantons, categorized by affine root systems, and their charged/neutral bions govern the non-perturbative dynamics, with neutral bions embodying IR renormalon effects and matching to the strong-scale generation via the beta-function ${\beta_0}=h^{\vee}$. For Grassmannians, 2d instantons fractionalize into $N$ kink-instantons, giving a universal pattern of one- and two-event dynamics and theta-angle multi-branching, while large-$N$ limits exhibit theta-angle independence. The study also draws a precise Lie-algebraic link between 2d saddles and monopole-instantons in gauge theories, illuminating a cohesive picture where resurgent structures connect perturbation theory to non-perturbative saddles and mass-gap formation, with clear implications for CP properties and observable Theta-angle dependencies.

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 $\mathbb R \times S^1$, 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 $\mathbb R \times S^1$, 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 $\mathbb R^3 \times S^1$. The Grassmannian sigma models ${ \rm Gr}(N, M)$ 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 ${ \rm Gr}(N, M)$, and describe the multi-branched structure of the observables in terms of the theta-angle dependence of the saddle amplitudes.

Figures (6)

• Figure 1: Twisted boundary condition for the $O(2M)$ (left) and $O(2M+1)$ (right) models. The blue points are mirror images of the red ones. $\pm \mu_1$ and $\pm\mu_M$ are coincident, but are split for convenience of visualization. For $O (2M+1)$, the eigenvalue at zero does not have a mirror image.
• Figure 2: Minimal action saddles associated with the affine root system of the simply-laced ${\frak o} (2M)$ algebra (top) and the root system of the non-simply laced ${\frak o} (2M+1)$ algebra (bottom). In the simply laced case, the action of the saddles are proportional to the eigenvalue differences. In the non-simply laced case, the short root (and its KK-tower) requires more care, as discussed in the text.
• Figure 3: Fig. \ref{['fig:TBC-instanton']} can be reinterpreted in terms of the Dynkin diagrams of $D_M=O(2M)$ and $B_M=O(2M+1)$. The shaded circles denote the affine affine roots, and are present because the theory is compactified on a circle. There is a short root in the non-simply-laced $O(2M+1)$ case. The above diagrams should be used for $B_{M \geq 3}=O((2M+1) \geq 7)$, and $D_{M \geq 4}=O((2M) \geq 8)$, with lower rank cases requiring slightly more care due to additional symmetries.
• Figure 4: A snap-shot of the dilute gas of one- and two-events, kink-saddles (associated with roots) and bions (associated with non-zero entries of the extended Cartan matrix), respectively. The amplitude of the neutral bions are two-fold ambiguous, fixing the ambiguity of perturbation theory.
• Figure 5: The $\Theta$ angle dependence of various observables: Left: spin wave condensate $O_1(\Theta)$ and mass gap $m_g (\Theta)$. Right: Topological chage density condensate $O_T(\Theta)$ in $O(3)$ model. Each is a two-branched function. $O_1(\Theta)$ has a cusp at $\Theta=\pi$ associated with a change of branch. $O_T(\Theta)$ has a discontinuity at $\Theta=\pi$.