Resurgence and dynamics of O(N) and Grassmannian sigma models

Gerald V Dunne,Mithat Ünsal

doi:10.1007/jhep09(2015)199

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 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 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 Gr(N, M ), and describe the multi-branched structure of the observables in terms of the theta-angle dependence of the saddle amplitudes, providing a microscopic argument for Haldane’s conjecture.

Highlights

We apply the resurgence formalism and the physical principle of adiabatic continuity on R×S1 to the simplest asymptotically free QFT with a mass gap, the O(N ) non-linear sigma model in two spacetime dimensions (2d)
We study the non-perturbative dynamics of the two dimensional O(N ) and Grassmannian sigma models by using compactification with twisted boundary conditions on R × S1, semi-classical techniques and resurgence
We show that the key to understanding this apparent puzzle is that, the O(N ) model with N ≥ 4 has no instantons, it has smooth finite action classical solutions of the second-order classical equations of motion [they are saddle points, not minima, of the action], and these play an important role in non-perturbative physics

Summary

Introduction

We apply the resurgence formalism and the physical principle of adiabatic continuity on R×S1 to the simplest asymptotically free QFT with a mass gap, the O(N ) non-linear sigma model in two spacetime dimensions (2d). For CPN−1 it was shown explicitly that the leading ambiguity in the Borel plane of perturbation theory [in the compactified theory] corresponds to neutral bions, the semi-classical realization of the IR-renormalons, and is cancelled by an ambiguity in non-perturbative bion amplitudes This provides an explicit realization of resurgence in a non-trivial QFT. This question was addressed in earlier work on the SU(N ) principal chiral model, which has no instantons, and where the 2d uniton saddle (a non-BPS solution to second order equations of motion) fractionates into N fracton constituents [5, 6, 33] This puzzle has remained unanswered for the O(N ≥ 4) vector model in 2d, the simplest asymptotically free QFT..

Objectives

Results

Conclusion