Resurgence and dynamics of O(N) and Grassmannian sigma models
Abstract
Here, we study the nonperturbative dynamics of the two dimensional O(N) and Grassmannian sigma models by using compactification with twisted boundary conditions on R×S^{1}, semiclassical techniques and resurgence. While the O(N) model has no instantons for N > 3, it has (noninstanton) saddles on R^{2}, which we call 2dsaddles. On R×S^{1}, the resurgent relation between perturbation theory and nonperturbative 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 2dsaddles. 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 onetoone correspondence with monopoleinstanton 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 kinkinstantons. The small circle dynamics of both sigma models can be described as a dilute gas of the oneevents and twoevents, bions. Oneevents are the leading source of a variety of nonperturbative effects, and produce the strong scale of the 2d theory in the compactified theory. We show thatmore »
 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 10298479
 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 nonperturbative dynamics of the two dimensional O(N) and Grassmannian sigma models by using compactification with twisted boundary conditions on R×S1, semiclassical techniques and resurgence. While the O(N) model has no instantons for N > 3, it has (noninstanton) saddles on R2, which we call 2dsaddles. On R×S1, the resurgent relation between perturbation theory and nonperturbative 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 2dsaddles. 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 onetoone correspondence with monopoleinstanton saddles in SO(N) gauge theory on R3×S1. The Grassmannian sigma models Gr(N, M) have 2d instantons, which fractionalize into N kinkinstantons. The small circle dynamics of both sigma models can be described as a dilute gas of the oneevents and twoevents, bions. Oneevents are the leading source of a variety of nonperturbative 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 IRrenormalons. We also study the topological theta angle dependence in both the O(3) model and Gr(N, M), and describe the multibranched structure of the observables in terms of the thetaangle 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}
}
Web of Science
Works referenced in this record:
Classical solutions in Grassmannian σmodels
journal, November 1981
 Din, A. M.; Zakrzewski, W. J.
 Letters in Mathematical Physics, Vol. 5, Issue 6
Solutions to YangMills equations that are not selfdual
journal, November 1989
 Sibner, L. M.; Sibner, R. J.; Uhlenbeck, K.
 Proceedings of the National Academy of Sciences, Vol. 86, Issue 22
Factorized Smatrices in two dimensions as the exact solutions of certain relativistic quantum field theory models
journal, August 1979
 Zamolodchikov, Alexander B.; Zamolodchikov, Alexey B.
 Annals of Physics, Vol. 120, Issue 2
θ dependence of SU(N) gauge theories in the presence of a topological term
journal, January 2009
 Vicari, Ettore; Panagopoulos, Haralambos
 Physics Reports, Vol. 470, Issue 34
Instanton Constituents in the O(3) Model at Finite Temperature
journal, February 2008
 Bruckmann, Falk
 Physical Review Letters, Vol. 100, Issue 5
Theta Dependence in the Large $\mathit{N}$ Limit of FourDimensional Gauge Theories
journal, October 1998
 Witten, Edward
 Physical Review Letters, Vol. 81, Issue 14
Comments on operators with large spin
journal, November 2007
 Alday, Luis F.; Maldacena, Juan
 Journal of High Energy Physics, Vol. 2007, Issue 11
On the ambiguity of composite operators, IR renormalons and the status of the operator product expansion
journal, March 1984
 David, F.
 Nuclear Physics B, Vol. 234, Issue 1
Lectures on nonperturbative effects in large N gauge theories, matrix models and strings: Nonperturbative effects in large N gauge theories, matrix models and strings
journal, April 2014
 Mariño, M.
 Fortschritte der Physik, Vol. 62, Issue 56
On the structure of infrared renormalons in physical processes at high energies
journal, January 1985
 Mueller, A. H.
 Nuclear Physics B, Vol. 250, Issue 14
Nonperturbative effects and the largeorder behavior of matrix models and topological strings
journal, January 2008
 Mariño, Marcos; Schiappa, Ricardo; Weiss, Marlene
 Communications in Number Theory and Physics, Vol. 2, Issue 2
Nonselfdual YangMills connections with quadrupole symmetry
journal, April 1992
 Sadun, Lorenzo; Segert, Jan
 Communications in Mathematical Physics, Vol. 145, Issue 2
The operator product expansion and renormalons: A comment
journal, January 1986
 David, F.
 Nuclear Physics B, Vol. 263, Issue 34
YangMills fields which are not selfdual
journal, April 1992
 Bor, Gil
 Communications in Mathematical Physics, Vol. 145, Issue 2
Semiclassical Realization of Infrared Renormalons
journal, September 2012
 Argyres, Philip; Ünsal, Mithat
 Physical Review Letters, Vol. 109, Issue 12
The exact mass gap of the O(N) σmodel for arbitrary N ⩾ 3 in d = 2
journal, August 1990
 Hasenfratz, P.; Niedermayer, F.
 Physics Letters B, Vol. 245, Issue 34
Neutral bions in the CP ^{N1} model for resurgence
journal, April 2015
 Misumi, Tatsuhiro; Nitta, Muneto; Sakai, Norisuke
 Journal of Physics: Conference Series, Vol. 597
Evading the sign problem in the meanfield approximation through Lefschetzthimble path integral
journal, May 2015
 Tanizaki, Yuya; Nishimura, Hiromichi; Kashiwa, Kouji
 Physical Review D, Vol. 91, Issue 10
Harmonic maps into Lie groups: classical solutions of the chiral model
journal, January 1989
 Uhlenbeck, Karen
 Journal of Differential Geometry, Vol. 30, Issue 1
Universal mechanism of (semiclassical) deconfinement and θdependence for all simple groups
journal, March 2013
 Poppitz, Erich; Schäfer, Thomas; Ünsal, Mithat
 Journal of High Energy Physics, Vol. 2013, Issue 3
Deconfinement and continuity between thermal and (super) YangMills theory for all gauge groups
journal, September 2014
 Anber, Mohamed M.; Poppitz, Erich; Teeple, Brett
 Journal of High Energy Physics, Vol. 2014, Issue 9
Generalizations of σmodels and CpN models, and instantons
journal, March 1979
 MacFarlane, A. J.
 Physics Letters B, Vol. 82, Issue 2
Theory of nonabelian goldstone bosons in two dimensions
journal, November 1983
 Polyakov, A.; Wiegmann, P. B.
 Physics Letters B, Vol. 131, Issue 13
Exact results in nonsupersymmetric large N orientifold field theories
journal, September 2003
 Armoni, A.; Shifman, M.; Veneziano, G.
 Nuclear Physics B, Vol. 667, Issue 12
Cusp Anomalous Dimension in Maximally Supersymmetric YangMills Theory at Strong Coupling
journal, March 2008
 Basso, B.; Korchemsky, G. P.; Kotański, J.
 Physical Review Letters, Vol. 100, Issue 9
Classical solutions of the chiral model, unitons, and holomorphic vector bundles
journal, March 1990
 Ward, R. S.
 Communications in Mathematical Physics, Vol. 128, Issue 2
Exact GellMannLow function of supersymmetric Kähler sigma models
journal, December 1984
 Morozov, Alexei Y.; Perelomov, Askold M.; Shifman, Michael A.
 Nuclear Physics B, Vol. 248, Issue 2
Fractional instantons and bions in the O(N) model with twisted boundary conditions
journal, March 2015
 Nitta, Muneto
 Journal of High Energy Physics, Vol. 2015, Issue 3
Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation
journal, August 2015
 Heller, Michal P.; Spaliński, Michał
 Physical Review Letters, Vol. 115, Issue 7
On finite action solutions of the nonlinear σmodel
journal, June 1979
 Garber, WolfDieter; Ruijsenaars, Simon N. M.; Seiler, Erhard
 Annals of Physics, Vol. 119, Issue 2
Hidden Topological Angles in Path Integrals
journal, July 2015
 Behtash, Alireza; Sulejmanpasic, Tin; Schäfer, Thomas
 Physical Review Letters, Vol. 115, Issue 4
Analyticity of solutions of theO(N) nonlinear ?model
journal, October 1980
 Borchers, H. J.; Garber, W. D.
 Communications in Mathematical Physics, Vol. 71, Issue 3
Nonlinear Field Theory of LargeSpin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the OneDimensional EasyAxis Néel State
journal, April 1983
 Haldane, F. D. M.
 Physical Review Letters, Vol. 50, Issue 15
Fractional instantons and bions in the principal chiral model on ℝ
2 × S
1 $$ {\mathrm{\mathbb{R}}}^2\times {S}^1 $$ with twisted boundary conditions
journal, August 2015
 Nitta, Muneto
 Journal of High Energy Physics, Vol. 2015, Issue 8
Classifying bions in Grassmann sigma models and nonAbelian gauge theories by Dbranes
journal, March 2015
 Misumi, Tatsuhiro; Nitta, Muneto; Sakai, Norisuke
 Progress of Theoretical and Experimental Physics, Vol. 2015, Issue 3
New and old about renormalons: In memoriam Kolya Uraltsev
journal, April 2015
 Shifman, M.
 International Journal of Modern Physics A, Vol. 30, Issue 10
Structure of Lefschetz thimbles in simple fermionic systems
journal, March 2015
 Kanazawa, Takuya; Tanizaki, Yuya
 Journal of High Energy Physics, Vol. 2015, Issue 3
Resurgence and transseries in Quantum Field Theory: the $\mathbb{C}{{\mathbb{P}}^{N1 }}$ model
journal, November 2012
 Dunne, Gerald V.; Ünsal, Mithat
 Journal of High Energy Physics, Vol. 2012, Issue 11
Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local $${\mathbb{C}\mathbb{P}^2}$$ C P 2
journal, April 2015
 CousoSantamaría, Ricardo; Edelstein, José D.; Schiappa, Ricardo
 Communications in Mathematical Physics, Vol. 338, Issue 1
Embedding nonlinear sigma model into superYang–Mills theory
journal, February 2009
 Basso, B.; Korchemsky, G. P.
 Nuclear Physics B, Vol. 807, Issue 3
Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models
journal, June 2010
 Pasquetti, Sara; Schiappa, Ricardo
 Annales Henri Poincaré, Vol. 11, Issue 3
The resurgence of instantons in string theory
journal, January 2012
 Aniceto, Inês; Schiappa, Ricardo; Vonk, Marcel
 Communications in Number Theory and Physics, Vol. 6, Issue 2
Nonperturbative equivalences among large N _{c} gauge theories with adjoint and bifundamental matter fields
journal, December 2003
 Kovtun, Pavel; Ünsal, Mithat; Yaffe, Laurence G.
 Journal of High Energy Physics, Vol. 2003, Issue 12
Stationary points of the YangMills action
journal, April 1992
 Sadun, Lorenzo; Segert, Jan
 Communications on Pure and Applied Mathematics, Vol. 45, Issue 4
Resurgence in Quantum Field Theory: Nonperturbative Effects in the Principal Chiral Model
journal, January 2014
 Cherman, Aleksey; Dorigoni, Daniele; Dunne, Gerald V.
 Physical Review Letters, Vol. 112, Issue 2
The exact mass gap of the O(3) and O(4) nonlinear σmodels in d = 2
journal, August 1990
 Hasenfratz, P.; Maggiore, M.; Niedermayer, F.
 Physics Letters B, Vol. 245, Issue 34
ChernSimons solitons, Toda theories and the chiral model
journal, December 1992
 Dunne, Gerald
 Communications in Mathematical Physics, Vol. 150, Issue 3
The semiclassical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects
journal, August 2012
 Argyres, Philip C.; Ünsal, Mithat
 Journal of High Energy Physics, Vol. 2012, Issue 8
Neutral bions in the ℂ $$ \mathbb{C} $$ P N −1 model
journal, June 2014
 Misumi, Tatsuhiro; Nitta, Muneto; Sakai, Norisuke
 Journal of High Energy Physics, Vol. 2014, Issue 6
Resurgent analysis of localizable observables in supersymmetric gauge theories
journal, March 2015
 Aniceto, Inês; Russo, Jorge G.; Schiappa, Ricardo
 Journal of High Energy Physics, Vol. 2015, Issue 3
Periodic instantons with nontrivial holonomy
journal, November 1998
 Kraan, Thomas C.; van Baal, Pierre
 Nuclear Physics B, Vol. 533, Issue 13
Embeddings of classical solutions of 02p+1 nonlinear Σmodels in CPn1 models
journal, May 1980
 Din, A. M.; Zakrzewski, W. J.
 Lettere Al Nuovo Cimento Series 2, Vol. 28, Issue 4
Wilson's operator expansion: Can it fail?
journal, January 1985
 Novikov, V. A.; Shifman, M. A.; Vainshtein, A. I.
 Nuclear Physics B, Vol. 249, Issue 3
Stability properties of classical solutions to nonlinear σ models
journal, May 1980
 Din, A. M.; Zakrzewski, W. J.
 Nuclear Physics B, Vol. 168, Issue 1
Threeloop correction to the instanton density. II. The sineGordon potential
journal, July 2015
 EscobarRuiz, M. A.; Shuryak, E.; Turbiner, A. V.
 Physical Review D, Vol. 92, Issue 2
Threeloop correction to the instanton density. I. The quartic double well potential
journal, July 2015
 EscobarRuiz, M. A.; Shuryak, E.; Turbiner, A. V.
 Physical Review D, Vol. 92, Issue 2
A note on instanton effects in ABJM theory
journal, November 2014
 Wang, Xianfu; Wang, Xin; Huang, Minxin
 Journal of High Energy Physics, Vol. 2014, Issue 11
Instanton constituents and fermionic zero modes in twisted <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchemainstance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/structbib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi mathvariant="doublestruck">C</mml:mi><mml:msup><mml:mi mathvariant="normal">P</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math> models
journal, June 2009
 Brendel, Wieland; Bruckmann, Falk; Janssen, Lukas
 Physics Letters B, Vol. 676, Issue 13
Nonperturbative effects in the exact Smatrices of the O(N) grossneveu and supersymmetric σ models
journal, December 1988
 Gracey, J. A.
 Physics Letters B, Vol. 215, Issue 3
Resurgence and the NekrasovShatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems
journal, February 2015
 Basar, Gökçe; Dunne, Gerald V.
 Journal of High Energy Physics, Vol. 2015, Issue 2
Necessary and sufficient conditions for nonperturbative equivalences of large N _{c} orbifold gauge theories
journal, July 2005
 Kovtun, Pavel; Ünsal, Mithat; Yaffe, Laurence G.
 Journal of High Energy Physics, Vol. 2005, Issue 07
From Weak to Strong Coupling in ABJM Theory
journal, May 2011
 Drukker, Nadav; Mariño, Marcos; Putrov, Pavel
 Communications in Mathematical Physics, Vol. 306, Issue 2
Works referencing / citing this record:
O(3) nonlinear sigma model in $1+1$ dimensions with matrix product states
journal, April 2019
 Bruckmann, Falk; Jansen, Karl; Kühn, Stefan
 Physical Review D, Vol. 99, Issue 7
O(3) nonlinear sigma model in $1+1$ dimensions with matrix product states
journal, April 2019
 Bruckmann, Falk; Jansen, Karl; Kühn, Stefan
 Physical Review D, Vol. 99, Issue 7
Bion nonperturbative contributions versus infrared renormalons in twodimensional ℂPN − 1 models
journal, February 2019
 Fujimori, Toshiaki; Kamata, Syo; Misumi, Tatsuhiro
 Journal of High Energy Physics, Vol. 2019, Issue 2
From 4d YangMills to 2d ℂℙN − 1 model: IR problem and confinement at weak coupling
journal, July 2017
 Yamazaki, Masahito; Yonekura, Kazuya
 Journal of High Energy Physics, Vol. 2017, Issue 7
Quantum distillation of Hilbert spaces, semiclassics and anomaly matching
journal, August 2018
 Dunne, Gerald V.; Tanizaki, Yuya; Ünsal, Mithat
 Journal of High Energy Physics, Vol. 2018, Issue 8
Renormalons in quantum mechanics
journal, August 2019
 Pazarbası, Cihan; Van den Bleeken, Dieter
 Journal of High Energy Physics, Vol. 2019, Issue 8
Resurgence for superconductors
journal, December 2019
 Mariño, Marcos; Reis, Tomás
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2019, Issue 12
Anomaly and global inconsistency matching: $\theta $ angles, $SU\left(3\right)/U{\left(1\right)}^{2}$ nonlinear sigma model, $SU\left(3\right)$ chains, and generalizations
journal, September 2018
 Tanizaki, Yuya; Sulejmanpasic, Tin
 Physical Review B, Vol. 98, Issue 11
Universal renormalons in principal chiral models
journal, February 2020
 Bruckmann, Falk; Puhr, Matthias
 Physical Review D, Vol. 101, Issue 3
Nonperturbative contributions from complexified solutions in $\mathbb{C}{P}^{N1}$ models
journal, November 2016
 Fujimori, Toshiaki; Kamata, Syo; Misumi, Tatsuhiro
 Physical Review D, Vol. 94, Issue 10
Exponential reduction of finite volume effects with twisted boundary conditions
journal, April 2017
 Cherman, Aleksey; Sen, Srimoyee; Wagman, Michael L.
 Physical Review D, Vol. 95, Issue 7
Global Symmetries, Volume Independence, and Continuity in Quantum Field Theories
journal, January 2017
 Sulejmanpasic, Tin
 Physical Review Letters, Vol. 118, Issue 1