Abstract
In this work we revisit the problem of the quantization of the two-dimensional O(3) non-linear sigma model and its one-parameter integrable deformation — the sausage model. Our consideration is based on the so-called ODE/IQFT correspondence, a variant of the Quantum Inverse Scattering Method. The approach allowed us to explore the integrable structures underlying the quantum O(3)/sausage model. Among the obtained results is a system of non-linear integral equations for the computation of the vacuum eigenvalues of the quantum transfer-matrices.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.D. Faddeev, E.K. Sklyanin and L.A. Takhtajan, The quantum inverse problem method. 1, Theor. Math. Phys. 40 (1980) 688 [Teor. Mat. Fiz. 40 (1979) 194] [INSPIRE].
R.J. Baxter, Partition function of the eight vertex lattice model, Annals Phys. 70 (1972) 193 [INSPIRE].
R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London U.K., (1982) [INSPIRE].
C. Destri and H.J. de Vega, Light cone lattice approach to fermionic theories in 2D: the massive Thirring model, Nucl. Phys. B 290 (1987) 363 [INSPIRE].
K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Commun. Math. Phys. 46 (1976) 207 [INSPIRE].
V.E. Zakharov and A.V. Mikhailov, Relativistically invariant two-dimensional models in field theory integrable by the inverse problem technique (in Russian), Sov. Phys. JETP 47 (1978) 1017 [Zh. Eksp. Teor. Fiz. 74 (1978) 1953] [INSPIRE].
A.M. Polyakov and P.B. Wiegmann, Theory of non-Abelian Goldstone bosons, Phys. Lett. B 131 (1983) 121 [INSPIRE].
L.D. Faddeev and N. Yu. Reshetikhin, Integrability of the principal chiral field model in (1 + 1)-dimension, Annals Phys. 167 (1986) 227 [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].
A.G. Bytsko, The zero curvature representation for nonlinear O(3) σ-model, J. Math. Sci. 85 (1994) 1619 [hep-th/9403101] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381 [hep-th/9412229] [INSPIRE].
D. Fioravanti and M. Rossi, A braided Yang-Baxter algebra in a theory of two coupled lattice quantum KdV: algebraic properties and ABA representations, J. Phys. A 35 (2002) 3647 [hep-th/0104002] [INSPIRE].
D. Fioravanti and M. Rossi, Exact conserved quantities on the cylinder 1: conformal case, JHEP 07 (2003) 031 [hep-th/0211094] [INSPIRE].
V.A. Fateev, E. Onofri and A.B. Zamolodchikov, The sausage model (integrable deformations of O(3) σ-model), Nucl. Phys. B 406 (1993) 521 [INSPIRE].
S.L. Lukyanov, The integrable harmonic map problem versus Ricci flow, Nucl. Phys. B 865 (2012) 308 [arXiv:1205.3201] [INSPIRE].
E.K. Sklyanin, On the complete integrability of the Landau-Lifshitz equation, LOMI-E-79-3, Russia, (1980).
L.D. Faddeev and L.A. Takhtajan, Hamiltonian methods in the theory of solitons, Springer-Verlag, Berlin Germany, (1987) [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory II. Q-operator and DDV equation, Commun. Math. Phys. 190 (1997) 247 [hep-th/9604044] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory III. The Yang-Baxter relation, Commun. Math. Phys. 200 (1999) 297 [hep-th/9805008] [INSPIRE].
A. Voros, Spectral zeta functions, Adv. Stud. Pure Math. 21 (1992) 327.
A. Voros, An exact solution method for 1D polynomial Schrödinger equations, J. Phys. A 32 (1999) 5993 [math-ph/9902016].
P. Dorey and R. Tateo, Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations, J. Phys. A 32 (1999) L419 [hep-th/9812211] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Spectral determinants for Schrödinger equation and Q-operators of conformal field theory, J. Statist. Phys. 102 (2001) 567 [hep-th/9812247] [INSPIRE].
J. Suzuki, Functional relations in Stokes multipliers and solvable models related to U q (A (1) n ), J. Phys. A 33 (2000) 3507 [hep-th/9910215] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Higher level eigenvalues of Q-operators and Schrödinger equation, Adv. Theor. Math. Phys. 7 (2003) 711 [hep-th/0307108] [INSPIRE].
Al.B. Zamolodchikov, Generalized Mathieu equation and Liouville TBA, in Quantum field theories in two dimensions, collected works of Alexei Zamolodchikov, volume 2, A. Belavin, Ya. Pugai and A. Zamolodchikov eds., World Scientific, Singapore, (2012).
S.L. Lukyanov and A.B. Zamolodchikov, Quantum sine(h)-Gordon model and classical integrable equations, JHEP 07 (2010) 008 [arXiv:1003.5333] [INSPIRE].
P. Dorey, C. Dunning and R. Tateo, The ODE/IM correspondence, J. Phys. A 40 (2007) R205 [hep-th/0703066] [INSPIRE].
P. Dorey, S. Faldella, S. Negro and R. Tateo, The Bethe ansatz and the Tzitzeica-Bullough-Dodd equation, Phil. Trans. Roy. Soc. Lond. A 371 (2013) 20120052 [arXiv:1209.5517] [INSPIRE].
P. Adamopoulou and C. Dunning, Bethe ansatz equations for the classical A (1) n affine Toda field theories, J. Phys. A 47 (2014) 205205 [arXiv:1401.1187] [INSPIRE].
K. Ito and C. Locke, ODE/IM correspondence and Bethe ansatz for affine Toda field equations, Nucl. Phys. B 896 (2015) 763 [arXiv:1502.00906] [INSPIRE].
D. Masoero, A. Raimondo and D. Valeri, Bethe ansatz and the spectral theory of affine Lie algebra-valued connections I. The simply-laced case, Commun. Math. Phys. 344 (2016) 719 [arXiv:1501.07421] [INSPIRE].
D. Masoero, A. Raimondo and D. Valeri, Bethe ansatz and the spectral theory of affine Lie algebra-valued connections II. The non simply-laced case, Commun. Math. Phys. 349 (2017) 1063 [arXiv:1511.00895] [INSPIRE].
K. Ito and H. Shu, ODE/IM correspondence for modified B (1)2 affine Toda field equation, Nucl. Phys. B 916 (2017) 414 [arXiv:1605.04668] [INSPIRE].
C. Babenko and F. Smirnov, Suzuki equations and integrals of motion for supersymmetric CFT, Nucl. Phys. B 924 (2017) 406 [arXiv:1706.03349] [INSPIRE].
R.S. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71, Amer. Math. Soc., Providence RI U.S.A., (1988), pg. 237.
S. Elitzur, A. Forge and E. Rabinovici, Some global aspects of string compactifications, Nucl. Phys. B 359 (1991) 581 [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].
Yu. G. Stroganov, A new calculation method for partition functions in some lattice models, Phys. Lett. A 74 (1979) 116 [INSPIRE].
R.J. Baxter and P.A. Pearce, Hard hexagons: interfacial tension and correlation length, J. Phys. A 15 (1982) 897.
A.N. Kirillov and N.Y. Reshetikhin, Exact solution of the integrable XXZ Heisenberg model with arbitrary spin I. The ground state and the excitation spectrum, J. Phys. A 20 (1987) 1565 [INSPIRE].
S.L. Lukyanov, E.S. Vitchev and A.B. Zamolodchikov, Integrable model of boundary interaction: the paperclip, Nucl. Phys. B 683 (2004) 423 [hep-th/0312168] [INSPIRE].
S.L. Lukyanov, A.M. Tsvelik and A.B. Zamolodchikov, Paperclip at θ = π, Nucl. Phys. B 719 (2005) 103 [hep-th/0501155] [INSPIRE].
S.L. Lukyanov and A.B. Zamolodchikov, Integrability in 2D field theory/sigma models, lecture notes for the Les Houches school, to appear, France, (2016).
I. Bakas and E. Kiritsis, Beyond the large-N limit: nonlinear W ∞ as symmetry of the SL(2, R)/U(1) coset model, Int. J. Mod. Phys. A 7S1A (1992) 55 [hep-th/9109029] [INSPIRE].
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, String propagation in a black hole geometry, Nucl. Phys. B 371 (1992) 269 [INSPIRE].
A.B. Zamolodchikov and Al.B. Zamolodchikov, unpublished notes, (1995).
A. Gerasimov, A. Marshakov and A. Morozov, Free field representation of parafermions and related coset models, Nucl. Phys. B 328 (1989) 664 [INSPIRE].
V.A. Fateev and A.B. Zamolodchikov, Parafermionic currents in the two-dimensional conformal quantum field theory and selfdual critical points in Z N invariant statistical systems, Sov. Phys. JETP 62 (1985) 215 [Zh. Eksp. Teor. Fiz. 89 (1985) 380] [INSPIRE].
V.A. Fateev and A.B. Zamolodchikov, Selfdual solutions of the star triangle relations in Z N models, Phys. Lett. A 92 (1982) 37 [INSPIRE].
G. Felder, BRST approach to minimal models, Nucl. Phys. B 317 (1989) 215 [Erratum ibid. B 324 (1989) 548] [INSPIRE].
A.G. Izergin and V.E. Korepin, The lattice quantum sine-Gordon model, Lett. Math. Phys. 5 (1981) 199 [INSPIRE].
E.K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras, Funct. Anal. Appl. 17 (1983) 273 [Funkt. Anal. Pril. 17 (1983) 34] [INSPIRE].
V.V. Bazhanov and Yu. G. Stroganov, Chiral Potts model as a descendant of the six vertex model, J. Statist. Phys. 59 (1990) 799 [INSPIRE].
R.J. Baxter, V.V. Bazhanov and J.H.H. Perk, Functional relations for transfer matrices of the chiral Potts model, Int. J. Mod. Phys. B 4 (1990) 803 [INSPIRE].
E.K. Sklyanin, Separation of variables — new trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [INSPIRE].
S.L. Lukyanov, ODE/IM correspondence for the Fateev model, JHEP 12 (2013) 012 [arXiv:1303.2566] [INSPIRE].
V.V. Bazhanov and S.L. Lukyanov, Integrable structure of quantum field theory: classical flat connections versus quantum stationary states, JHEP 09 (2014) 147 [arXiv:1310.4390] [INSPIRE].
D. Fioravanti, Geometrical loci and CFTs via the Virasoro symmetry of the mKdV-SG hierarchy: an excursus, Phys. Lett. B 609 (2005) 173 [hep-th/0408079] [INSPIRE].
B. Feigin and E. Frenkel, Quantization of soliton systems and Langlands duality, Adv. Stud. Pure Math. 61 (2011) 185 [arXiv:0705.2486] [INSPIRE].
A. Klumper, M.T. Batchelor and P.A. Pearce, Central charges of the 6- and 19-vertex models with twisted boundary conditions, J. Phys. A 24 (1991) 3111 [INSPIRE].
C. Destri and H.J. de Vega, New approach to thermal Bethe ansatz, Phys. Rev. Lett. 69 (1992) 2313 [hep-th/9203064] [INSPIRE].
C. Destri and H.J. De Vega, Unified approach to thermodynamic Bethe ansatz and finite size corrections for lattice models and field theories, Nucl. Phys. B 438 (1995) 413 [hep-th/9407117] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
V.A. Fateev, I.V. Frolov and A.S. Shvarts, Quantum fluctuations of instantons in the nonlinear σ-model, Nucl. Phys. B 154 (1979) 1 [INSPIRE].
A.P. Bukhvostov and L.N. Lipatov, Instanton-anti-instanton interaction in the O(3) nonlinear σ model and an exactly soluble fermion theory, Nucl. Phys. B 180 (1981) 116 [Pisma Zh. Eksp. Teor. Fiz. 31 (1980) 138] [INSPIRE].
M. Lüscher, Does the topological susceptibility in lattice σ-models scale according to the perturbative renormalization group?, Nucl. Phys. B 200 (1982) 61 [INSPIRE].
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories I. Stable particle states, Commun. Math. Phys. 104 (1986) 177 [INSPIRE].
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories II. Scattering states, Commun. Math. Phys. 105 (1986) 153 [INSPIRE].
C. Ahn, J. Balog and F. Ravanini, NLIE for the sausage model, arXiv:1701.08933 [INSPIRE].
A.M. Polyakov, Interaction of Goldstone particles in two-dimensions. Applications to ferromagnets and massive Yang-Mills fields, Phys. Lett. B 59 (1975) 79 [INSPIRE].
S. Hikami and E. Brézin, Three loop calculations in the two-dimensional nonlinear σ-model, J. Phys. A 11 (1978) 1141 [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].
M. Lüscher, P. Weisz and U. Wolff, A numerical method to compute the running coupling in asymptotically free theories, Nucl. Phys. B 359 (1991) 221 [INSPIRE].
D.-S. Shin, A determination of the mass gap in the O(N ) σ-model, Nucl. Phys. B 496 (1997) 408 [hep-lat/9611006] [INSPIRE].
J. Balog and A. Hegedus, The finite size spectrum of the 2-dimensional O(3) nonlinear σ-model, Nucl. Phys. B 829 (2010) 425 [arXiv:0907.1759] [INSPIRE].
G.V. Dunne and M. Ünsal, Resurgence and trans-series in quantum field theory: the CP N −1 model, JHEP 11 (2012) 170 [arXiv:1210.2423] [INSPIRE].
A. Dabholkar, Strings on a cone and black hole entropy, Nucl. Phys. B 439 (1995) 650 [hep-th/9408098] [INSPIRE].
V.A. Fateev and A.B. Zamolodchikov, Integrable perturbations of Z N parafermion models and O(3) σ-model, Phys. Lett. B 271 (1991) 91 [INSPIRE].
V.A. Fateev, The σ-model (dual) representation for a two-parameter family of integrable quantum field theories, Nucl. Phys. B 473 (1996) 509 [INSPIRE].
V.V. Bazhanov, G.A. Kotousov and S.L. Lukyanov, Winding vacuum energies in a deformed O(4) σ-model, Nucl. Phys. B 889 (2014) 817 [arXiv:1409.0449] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and B.A. Runov, Vacuum energy of the Bukhvostov-Lipatov model, Nucl. Phys. B 911 (2016) 863 [arXiv:1607.04839] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and B.A. Runov, Bukhvostov-Lipatov model and quantum-classical duality, arXiv:1711.09021 [INSPIRE].
J.M. Maillet, New integrable canonical structures in two-dimensional models, Nucl. Phys. B 269 (1986) 54 [INSPIRE].
F.A. Smirnov, Quasiclassical study of form-factors in finite volume, hep-th/9802132 [INSPIRE].
S.L. Lukyanov, Finite temperature expectation values of local fields in the sinh-Gordon model, Nucl. Phys. B 612 (2001) 391 [hep-th/0005027] [INSPIRE].
A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equation in sinh-Gordon model, J. Phys. A 39 (2006) 12863 [hep-th/0005181] [INSPIRE].
A.G. Bytsko and J. Teschner, Quantization of models with non-compact quantum group symmetry: modular XXZ magnet and lattice sinh-Gordon model, J. Phys. A 39 (2006) 12927 [hep-th/0602093] [INSPIRE].
J. Teschner, On the spectrum of the sinh-Gordon model in finite volume, Nucl. Phys. B 799 (2008) 403 [hep-th/0702214] [INSPIRE].
G. Borot, A. Guionnet and K.K. Kozlowski, Asymptotic expansion of a partition function related to the sinh-model, arXiv:1412.7721 [INSPIRE].
Y. Ikhlef, J.L. Jacobsen and H. Saleur, An integrable spin chain for the SL(2, R)/U(1) black hole σ-model, Phys. Rev. Lett. 108 (2012) 081601 [arXiv:1109.1119] [INSPIRE].
H. Boos, M. Jimbo, T. Miwa, F. Smirnov and Y. Takeyama, Hidden Grassmann structure in the XXZ model, Commun. Math. Phys. 272 (2007) 263 [hep-th/0606280] [INSPIRE].
H. Boos, M. Jimbo, T. Miwa, F. Smirnov and Y. Takeyama, Hidden Grassmann structure in the XXZ model II: creation operators, Commun. Math. Phys. 286 (2009) 875 [arXiv:0801.1176] [INSPIRE].
M. Jimbo, T. Miwa and F. Smirnov, Hidden Grassmann structure in the XXZ model III: introducing Matsubara direction, J. Phys. A 42 (2009) 304018 [arXiv:0811.0439] [INSPIRE].
H. Boos, M. Jimbo, T. Miwa and F. Smirnov, Hidden Grassmann structure in the XXZ model IV: CFT limit, Commun. Math. Phys. 299 (2010) 825 [arXiv:0911.3731] [INSPIRE].
M. Jimbo, T. Miwa and F. Smirnov, Hidden Grassmann structure in the XXZ model V: sine-Gordon model, Lett. Math. Phys. 96 (2011) 325 [arXiv:1007.0556] [INSPIRE].
P.B. Wiegmann, On the theory of non-Abelian Goldstone bosons in two-dimensions: exact solution of the O(3) nonlinear σ model, Phys. Lett. B 141 (1984) 217 [INSPIRE].
J. Balog and A. Hegedus, TBA equations for excited states in the O(3) and O(4) nonlinear σ-model, J. Phys. A 37 (2004) 1881 [hep-th/0309009] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from Hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Massless factorized scattering and σ-models with topological terms, Nucl. Phys. B 379 (1992) 602 [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum toroidal \( \mathfrak{g}{\mathfrak{l}}_1 \) and Bethe ansatz, J. Phys. A 48 (2015) 244001 [arXiv:1502.07194] [INSPIRE].
M.N. Alfimov and A.V. Litvinov, On spectrum of ILW hierarchy in conformal field theory II: coset CFT’s, JHEP 02 (2015) 150 [arXiv:1411.3313] [INSPIRE].
B. Feigin, M. Jimbo and E. Mukhin, Integrals of motion from quantum toroidal algebras, J. Phys. A 50 (2017) 464001 [arXiv:1705.07984].
C. Klimčík, Integrability of the bi-Yang-Baxter σ-model, Lett. Math. Phys. 104 (2014) 1095 [arXiv:1402.2105] [INSPIRE].
B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdS n × S n supercosets, JHEP 06 (2014) 002 [arXiv:1403.5517] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, Integrable double deformation of the principal chiral model, Nucl. Phys. B 891 (2015) 312 [arXiv:1410.8066] [INSPIRE].
K. Sfetsos, K. Siampos and D.C. Thompson, Generalised integrable λ- and η-deformations and their relation, Nucl. Phys. B 899 (2015) 489 [arXiv:1506.05784] [INSPIRE].
C. Klimčík, Poisson-Lie T-duals of the bi-Yang-Baxter models, Phys. Lett. B 760 (2016) 345 [arXiv:1606.03016] [INSPIRE].
A. Litvinov and L. Spodyneiko, On W algebras commuting with a set of screenings, JHEP 11 (2016) 138 [arXiv:1609.06271] [INSPIRE].
V. Fateev, A. Litvinov and L. Spodyneiko, private communications, to be published.
S.L. Lukyanov and A.B. Zamolodchikov, Integrable circular brane model and Coulomb charging at large conduction, J. Stat. Mech. 0405 (2004) P05003 [hep-th/0306188] [INSPIRE].
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
K.B. Efetov, Supersymmetry in disorder and chaos, Cambridge University Press, New York U.S.A., (1997).
G. Albertini, Bethe-ansatz type equations for the Fateev-Zamolodchikov spin model, J. Phys. A 25 (1992) 1799.
S. Ray, Bethe ansatz study for ground state of Fateev-Zamolodchikov model, J. Math. Phys. 38 (1997) 1524.
A.B. Zamolodchikov and V.A. Fateev, Model factorized S matrix and an integrable Heisenberg chain with spin 1 (in Russian), Sov. J. Nucl. Phys. 32 (1980) 298 [Yad. Fiz. 32 (1980) 581] [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: 1706.09941
To the memory of Ludwig Dmitrievich Faddeev.
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
Bazhanov, V.V., Kotousov, G.A. & Lukyanov, S.L. Quantum transfer-matrices for the sausage model. J. High Energ. Phys. 2018, 21 (2018). https://doi.org/10.1007/JHEP01(2018)021
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2018)021