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

## Abstract

Here, 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× *S* ^{1}, semi-classical techniques and resurgence. While the O(N) model has no instantons for N > 3, it has (non-instanton) saddles on R ^{2}, which we call 2d-saddles. On R× *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 R ^{3}×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 strongmore »

- Authors:

- Univ. of Connecticut, Storrs, CT (United States)
- North Carolina State Univ., Raleigh, NC (United States)

- Publication Date:

- Research Org.:
- Univ. of Connecticut, Storrs, CT (United States); North Carolina State Univ., Raleigh, NC (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1441165

- Grant/Contract Number:
- SC0010339; SC0013036

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2015; Journal Issue: 9; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Nonperturbative Effects; Field Theories in Lower Dimensions; Sigma Models

### Citation Formats

```
Dunne, Gerald V., and Unsal, Mithat. Resurgence and dynamics of O(N) and Grassmannian sigma models. United States: N. p., 2015.
Web. doi:10.1007/JHEP09(2015)199.
```

```
Dunne, Gerald V., & Unsal, Mithat. Resurgence and dynamics of O(N) and Grassmannian sigma models. United States. doi:10.1007/JHEP09(2015)199.
```

```
Dunne, Gerald V., and Unsal, Mithat. Tue .
"Resurgence and dynamics of O(N) and Grassmannian sigma models". United States. doi:10.1007/JHEP09(2015)199. https://www.osti.gov/servlets/purl/1441165.
```

```
@article{osti_1441165,
```

title = {Resurgence and dynamics of O(N) and Grassmannian sigma models},

author = {Dunne, Gerald V. and Unsal, Mithat},

abstractNote = {Here, 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. While the O(N) model has no instantons for N > 3, it has (non-instanton) saddles on R2, which we call 2d-saddles. On R×S1, 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 R3×S1. 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.},

doi = {10.1007/JHEP09(2015)199},

journal = {Journal of High Energy Physics (Online)},

number = 9,

volume = 2015,

place = {United States},

year = {2015},

month = {9}

}

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