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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P.C. Argyres and M. Ünsal, A semiclassical realization of infrared renormalons, Phys. Rev. Lett. 109 (2012) 121601 [arXiv:1204.1661] [INSPIRE].
P.C. Argyres and M. Ünsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion and renormalon effects, JHEP 08 (2012) 063 [arXiv:1206.1890] [INSPIRE].
G.V. Dunne and M. Ünsal, Resurgence and Trans-series in Quantum Field Theory: The \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) Model, JHEP 11 (2012) 170 [arXiv:1210.2423] [INSPIRE].
G.V. Dunne and M. Ünsal, Continuity and Resurgence: towards a continuum definition of the \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) model, Phys. Rev. D 87 (2013) 025015 [arXiv:1210.3646] [INSPIRE].
A. Cherman, D. Dorigoni, G.V. Dunne and M. Ünsal, Resurgence in Quantum Field Theory: Nonperturbative Effects in the Principal Chiral Model, Phys. Rev. Lett. 112 (2014) 021601 [arXiv:1308.0127] [INSPIRE].
A. Cherman, D. Dorigoni and M. Ünsal, Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles, arXiv:1403.1277 [INSPIRE].
T. Misumi, M. Nitta and N. Sakai, Neutral bions in the \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) model, JHEP 06 (2014) 164 [arXiv:1404.7225] [INSPIRE].
T. Misumi, M. Nitta and N. Sakai, Neutral bions in the \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) model for resurgence, J. Phys. Conf. Ser. 597 (2015) 012060 [arXiv:1412.0861] [INSPIRE].
G.V. Dunne and M. Ünsal, Generating nonperturbative physics from perturbation theory, Phys. Rev. D 89 (2014) 041701 [arXiv:1306.4405] [INSPIRE].
G.V. Dunne and M. Ünsal, Uniform WKB, Multi-instantons and Resurgent Trans-Series, Phys. Rev. D 89 (2014) 105009 [arXiv:1401.5202] [INSPIRE].
M.A. Escobar-Ruiz, E. Shuryak and A.V. Turbiner, Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential, Phys. Rev. D 92 (2015) 025046 [arXiv:1501.03993] [INSPIRE].
M.A. Escobar-Ruiz, E. Shuryak and A.V. Turbiner, Three-loop Correction to the Instanton Density. II. The sine-Gordon potential, Phys. Rev. D 92 (2015) 025047 [arXiv:1505.05115] [INSPIRE].
O. Costin, Asymptotics and Borel Summability, Chapman & Hall/CRC (2009).
E. Delabaere, Introduction to the Ecalle theory, in Computer Algebra and Differential Equations, Cambridge University Press (1994) [London Math. Soc. Lecture Note Ser. 193 (1994) 59].
D. Sauzin, Introduction to 1-summability and resurgence, arXiv:1405.0356.
M. Mariño, R. Schiappa and M. Weiss, Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings, Commun. Num. Theor. Phys. 2 (2008) 349 [arXiv:0711.1954] [INSPIRE].
S. Pasquetti and R. Schiappa, Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models, Ann. Henri Poincaré 11 (2010) 351 [arXiv:0907.4082] [INSPIRE].
I. Aniceto, R. Schiappa and M. Vonk, The Resurgence of Instantons in String Theory, Commun. Num. Theor. Phys. 6 (2012) 339 [arXiv:1106.5922] [INSPIRE].
M. Mariño, Lectures on non-perturbative effects in large-N gauge theories, matrix models and strings, Fortsch. Phys. 62 (2014) 455 [arXiv:1206.6272] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, arXiv:1308.6485 [INSPIRE].
J. Kallen, The spectral problem of the ABJ Fermi gas, arXiv:1407.0625 [INSPIRE].
X.-f. Wang, X. Wang and M.-x. Huang, A Note on Instanton Effects in ABJM Theory, JHEP 11 (2014) 100 [arXiv:1409.4967] [INSPIRE].
A. Grassi, Y. Hatsuda and M. Mariño, Quantization conditions and functional equations in ABJ(M) theories, arXiv:1410.7658 [INSPIRE].
R.C. Santamaría, J.D. Edelstein, R. Schiappa and M. Vonk, Resurgent Transseries and the Holomorphic Anomaly, arXiv:1308.1695 [INSPIRE].
R. Couso-Santamaría, J.D. Edelstein, R. Schiappa and M. Vonk, Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local \( \mathbb{C}{\mathrm{\mathbb{P}}}^2 \), Commun. Math. Phys. 338 (2015) 285 [arXiv:1407.4821] [INSPIRE].
I. Aniceto, J.G. Russo and R. Schiappa, Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories, JHEP 03 (2015) 172 [arXiv:1410.5834] [INSPIRE].
R. Dabrowski and G.V. Dunne, Fractionalized Non-Self-Dual Solutions in the \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) Model, Phys. Rev. D 88 (2013) 025020 [arXiv:1306.0921] [INSPIRE].
S. Bolognesi and W. Zakrzewski, Clustering and decomposition for non-BPS solutions of the \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) models, Phys. Rev. D 89 (2014) 065013 [arXiv:1310.8247] [INSPIRE].
F. Bruckmann and T. Sulejmanpasic, Nonlinear σ-models at nonzero chemical potential: breaking up instantons and the phase diagram, Phys. Rev. D 90 (2014) 105010 [arXiv:1408.2229] [INSPIRE].
T. Misumi, M. Nitta and N. Sakai, Classifying bions in Grassmann σ-models and non-Abelian gauge theories by D-branes, Prog. Theor. Exp. Phys. 2015 (2015) 033B02 [arXiv:1409.3444] [INSPIRE].
M. Nitta, Fractional instantons and bions in the O(N ) model with twisted boundary conditions, JHEP 03 (2015) 108 [arXiv:1412.7681] [INSPIRE].
M. Nitta, Fractional instantons and bions in the principal chiral model on \( {\mathbb{R}}^2\times {S}^1 \) with twisted boundary conditions, JHEP 08 (2015) 063 [arXiv:1503.06336] [INSPIRE].
Y. Tanizaki, Lefschetz-thimble techniques for path integral of zero-dimensional O(n) sigma-models, Phys. Rev. D 91 (2015) 036002 [arXiv:1412.1891] [INSPIRE].
T. Kanazawa and Y. Tanizaki, Structure of Lefschetz thimbles in simple fermionic systems, JHEP 03 (2015) 044 [arXiv:1412.2802] [INSPIRE].
Y. Tanizaki, H. Nishimura and K. Kashiwa, Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral, Phys. Rev. D 91 (2015) 101701(R) [arXiv:1504.02979] [INSPIRE].
G. Basar and G.V. Dunne, Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems, JHEP 02 (2015) 160 [arXiv:1501.05671] [INSPIRE].
M.P. Heller and M. Spalinski, Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation, Phys. Rev. Lett. 115 (2015) 072501 [arXiv:1503.07514] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized S-Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models, Annals Phys. 120 (1979) 253 [INSPIRE].
V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Wilson’s Operator Expansion: Can It Fail?, Nucl. Phys. B 249 (1985) 445 [Yad. Fiz. 41 (1985) 1063] [INSPIRE].
V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Two-Dimensional σ-models: Modeling Nonperturbative Effects of Quantum Chromodynamics, Phys. Rept. 116 (1984) 103 [Sov. J. Part. Nucl. 17 (1986) 204] [Fiz. Elem. Chast. Atom. Yadra 17 (1986) 472] [INSPIRE].
A.M. Polyakov and P.B. Wiegmann, Theory of Nonabelian Goldstone Bosons, Phys. Lett. B 131 (1983) 121 [INSPIRE].
J.A. Gracey, Non-perturbative effects in the exact S-matrices of the O(N ) gross-neveu and supersymmetric σ models, Phys. Lett. B 215 (1988) 505 [INSPIRE].
P. Hasenfratz, M. Maggiore and F. Niedermayer, The Exact mass gap of the O(3) and O(4) nonlinear σ-models in D = 2, Phys. Lett. B 245 (1990) 522 [INSPIRE].
P. Hasenfratz and F. Niedermayer, The Exact mass gap of the O(N ) σ-model for arbitrary N ≥3 in d = 2, Phys. Lett. B 245 (1990) 529 [INSPIRE].
D. Volin, From the mass gap in O(N ) to the non-Borel-summability in O(3) and O(4) σ-models, Phys. Rev. D 81 (2010) 105008 [arXiv:0904.2744] [INSPIRE].
M. Beneke, V.M. Braun and N. Kivel, The Operator product expansion, nonperturbative couplings and the Landau pole: Lessons from the O(N ) σ-model, Phys. Lett. B 443 (1998) 308 [hep-ph/9809287] [INSPIRE].
E. Abdalla, M. Abdalla and K. Rothe, Non-perturbative Methods in Two-Dimensional Quantum Field Theory, World Scientific, Singapore (2001).
W.J. Zakrzewski, Low Dimensional Sigma Models, CRC Press (1989).
F. David, The Operator Product Expansion and Renormalons: A Comment, Nucl. Phys. B 263 (1986) 637 [INSPIRE].
A. Armoni, M. Shifman and G. Veneziano, Exact results in nonsupersymmetric large-N orientifold field theories, Nucl. Phys. B 667 (2003) 170 [hep-th/0302163] [INSPIRE].
P. Kovtun, M. Ünsal and L.G. Yaffe, Necessary and sufficient conditions for non-perturbative equivalences of large-Nc orbifold gauge theories, JHEP 07 (2005) 008 [hep-th/0411177] [INSPIRE].
P. Kovtun, M. Ünsal and L.G. Yaffe, Nonperturbative equivalences among large-Nc gauge theories with adjoint and bifundamental matter fields, JHEP 12 (2003) 034 [hep-th/0311098] [INSPIRE].
F. David, On the Ambiguity of Composite Operators, IR Renormalons and the Status of the Operator Product Expansion, Nucl. Phys. B 234 (1984) 237 [INSPIRE].
M. Ünsal, Theta dependence, sign problems and topological interference, Phys. Rev. D 86 (2012) 105012 [arXiv:1201.6426] [INSPIRE].
A. Bhoonah, E. Thomas and A.R. Zhitnitsky, Metastable vacuum decay and θ dependence in gauge theory. Deformed QCD as a toy model, Nucl. Phys. B 890 (2014) 30 [arXiv:1407.5121] [INSPIRE].
M.M. Anber, Θ dependence of the deconfining phase transition in pure SU(Nc) Yang-Mills theories, Phys. Rev. D 88 (2013) 085003 [arXiv:1302.2641] [INSPIRE].
E. Poppitz, T. Schäfer and M. Ünsal, Universal mechanism of (semi-classical) deconfinement and θ-dependence for all simple groups, JHEP 03 (2013) 087 [arXiv:1212.1238] [INSPIRE].
M.M. Anber, E. Poppitz and B. Teeple, Deconfinement and continuity between thermal and (super) Yang-Mills theory for all gauge groups, JHEP 09 (2014) 040 [arXiv:1406.1199] [INSPIRE].
E. Witten, Large-N Chiral Dynamics, Annals Phys. 128 (1980) 363 [INSPIRE].
E. Witten, Theta dependence in the large-N limit of four-dimensional gauge theories, Phys. Rev. Lett. 81 (1998) 2862 [hep-th/9807109] [INSPIRE].
E. Vicari and H. Panagopoulos, θ dependence of SU(N ) gauge theories in the presence of a topological term, Phys. Rept. 470 (2009) 93 [arXiv:0803.1593] [INSPIRE].
M.E. Peskin and D.V. Schroeder, An Introduction to quantum field theory, Addison-Wesley, Reading U.S.A. (1995).
A.Yu. Morozov, A.M. Perelomov and M.A. Shifman, Exact Gell-Mann-Low function of supersymmetric Kähler sigma models, Nucl. Phys. B 248 (1984) 279 [INSPIRE].
J.L.M. Barbosa, On Minimal Immersions of S2 into S2m, Trans. Am. Math. Soc. 210 (1975) 75.
W.-D. Garber, S.N.M. Ruijsenaars and E. Seiler, On Finite Action Solutions of the Nonlinear σ Model, Annals Phys. 119 (1979) 305 [INSPIRE].
H.J. Borchers and W.-D. Garber, Analyticity of solutions of the O(N ) nonlinear σ-model, Commun. Math. Phys. 71 (1980) 299 [INSPIRE].
A.M. Din and W.J. Zakrzewski, Stability Properties of Classical Solutions to Nonlinear σ Models, Nucl. Phys. B 168 (1980) 173 [INSPIRE].
A.M. Din and W.J. Zakrzewski, Embeddings of Classical Solutions of 02p+1 Nonlinear σ Models in \( \mathbb{C}{\mathrm{\mathbb{P}}}^{n-1} \) Models, Lett. Nuovo Cim. 28 (1980) 121 [INSPIRE].
K.-M. Lee and P. Yi, Monopoles and instantons on partially compactified D-branes, Phys. Rev. D 56 (1997) 3711 [hep-th/9702107] [INSPIRE].
T.C. Kraan and P. van Baal, Periodic instantons with nontrivial holonomy, Nucl. Phys. B 533 (1998) 627 [hep-th/9805168] [INSPIRE].
J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge University Press (2000).
K. Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Diff. Geom. 30 (1989) 1.
R.S. Ward, Classical solutions of the chiral model, unitons and holomorphic vector bundles, Commun. Math. Phys. 128 (1990) 319 [INSPIRE].
G.V. Dunne, Chern-Simons solitons, Toda theories and the chiral model, Commun. Math. Phys. 150 (1992) 519 [hep-th/9204056] [INSPIRE].
G. ’t Hooft, Can We Make Sense Out of Quantum Chromodynamics?, in The Whys Of Subnuclear Physics, A. Zichichi ed., Plenum, New York U.S.A. (1979), p. 943.
G. Parisi, Singularities of the Borel Transform in Renormalizable Theories, Phys. Lett. B 76 (1978) 65 [INSPIRE].
G. Parisi, On Infrared Divergences, Nucl. Phys. B 150 (1979) 163 [INSPIRE].
A.H. Mueller, On the Structure of Infrared Renormalons in Physical Processes at High-Energies, Nucl. Phys. B 250 (1985) 327 [INSPIRE].
M. Beneke, Renormalons, Phys. Rept. 317 (1999) 1 [hep-ph/9807443] [INSPIRE].
M. Beneke and V.M. Braun, Renormalons and power corrections, in At the Frontier of Particle Physics. Vol. 3, M. Shifman ed., World Scientific, Singapore (2001), p. 1719 [hep-ph/0010208] [INSPIRE].
M. Shifman, New and Old about Renormalons: in Memoriam Kolya Uraltsev, Int. J. Mod. Phys. A 30 (2015) 1543001 [arXiv:1310.1966] [INSPIRE].
F. Bruckmann, Instanton constituents in the O(3) model at finite temperature, Phys. Rev. Lett. 100 (2008) 051602 [arXiv:0707.0775] [INSPIRE].
W. Brendel, F. Bruckmann, L. Janssen, A. Wipf and C. Wozar, Instanton constituents and fermionic zero modes in twisted \( \mathbb{C}{\mathrm{\mathbb{P}}}^n \) models, Phys. Lett. B 676 (2009) 116 [arXiv:0902.2328] [INSPIRE].
M. Eto et al., Non-Abelian vortices on cylinder: Duality between vortices and walls, Phys. Rev. D 73 (2006) 085008 [hep-th/0601181] [INSPIRE].
F.D.M. Haldane, Nonlinear field theory of large spin Heisenberg antiferromagnets. Semiclassically quantized solitons of the one-dimensional easy Axis Neel state, Phys. Rev. Lett. 50 (1983) 1153 [INSPIRE].
A.J. Macfarlane, Generalizations of σ Models and C N p Models and Instantons, Phys. Lett. B 82 (1979) 239 [INSPIRE].
A.M. Din and W.J. Zakrzewski, Classical Solutions in Grassmannian σ Models, Lett. Math. Phys. 5 (1981) 553 [INSPIRE].
J. Burzlaff, Nonselfdual Solutions of SU(3) Yang-Mills Theory and a Two-dimensional Abelian Higgs Model, Phys. Rev. D 24 (1981) 546 [INSPIRE].
L.M. Sibner, R.J. Sibner and K. Uhlenbeck, Solutions to Yang-Mills equations that are not self-dual, Proc. Natl. Acad. Sci. U.S.A. 86 (1989) 8610.
L. Sadun and J. Segert, Stationary Points of the Yang-Mills Action, Comm. Pure and Appl. Math. 45 (1992) 461
L. Sadun and J. Segert, Nonselfdual Yang-Mills connections with quadrupole symmetry, Commun. Math. Phys. 145 (1992) 363 [INSPIRE].
G. Bor, Yang-Mills fields which are not selfdual, Commun. Math. Phys. 145 (1992) 393 [INSPIRE].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP 11 (2007) 019 [arXiv:0708.0672] [INSPIRE].
B. Basso, G.P. Korchemsky and J. Kotanski, Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory at strong coupling, Phys. Rev. Lett. 100 (2008) 091601 [arXiv:0708.3933] [INSPIRE].
B. Basso and G.P. Korchemsky, Embedding nonlinear O(6) σ-model into N = 4 super-Yang-Mills theory, Nucl. Phys. B 807 (2009) 397 [arXiv:0805.4194] [INSPIRE].
A. Behtash, T. Sulejmanpasic, T. Schäfer and M. Ünsal, Hidden Topological Angles in Path Integrals, Phys. Rev. Lett. 115 (2015) 041601 [arXiv:1502.06624] [INSPIRE].
D. Dorigoni and Y. Hatsuda, Resurgence of the Cusp Anomalous Dimension, arXiv:1506.03763 [INSPIRE].
I. Aniceto, The Resurgence of the Cusp Anomalous Dimension, arXiv:1506.03388 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1505.07803
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Dunne, G.V., Ünsal, M. Resurgence and dynamics of O(N) and Grassmannian sigma models. J. High Energ. Phys. 2015, 199 (2015). https://doi.org/10.1007/JHEP09(2015)199
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2015)199