Abstract
We construct new integrable systems describing particles with internal spin from four-dimensional \( \mathcal{N} \) = 2 quiver gauge theories. The models can be quantized and solved exactly using the quantum inverse scattering method and also using the Bethe/Gauge correspondence.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [INSPIRE].
E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in proceedings of the 16th International Congress on Mathematical Physics (ICMP09), Prague, Czech Republic, 3-8 August 2009 [arXiv:0908.4052] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Bethe/Gauge correspondence on curved spaces, JHEP 01 (2015) 100 [arXiv:1405.6046] [INSPIRE].
D. Orlando and S. Reffert, The Gauge-Bethe Correspondence and Geometric Representation Theory, Lett. Math. Phys. 98 (2011) 289 [arXiv:1011.6120] [INSPIRE].
S. Hellerman, D. Orlando and S. Reffert, String theory of the Omega deformation, JHEP 01 (2012) 148 [arXiv:1106.0279] [INSPIRE].
D. Orlando and S. Reffert, Twisted Masses and Enhanced Symmetries: the A&D Series, JHEP 02 (2012) 060 [arXiv:1111.4811] [INSPIRE].
S. Hellerman, D. Orlando and S. Reffert, The omega deformation from string and M-theory, JHEP 07 (2012) 061 [arXiv:1204.4192] [INSPIRE].
K. Muneyuki, T.-S. Tai, N. Yonezawa and R. Yoshioka, Baxter’s T-Q equation, SU(N )/ SU(2)N −3 correspondence and Ω-deformed Seiberg-Witten prepotential, JHEP 09 (2011) 125 [arXiv:1107.3756] [INSPIRE].
A. Mironov, A. Morozov, Y. Zenkevich and A. Zotov, Spectral Duality in Integrable Systems from AGT Conjecture, JETP Lett. 97 (2013) 45 [Pisma Zh. Eksp. Teor. Fiz. 97 (2013) 49] [arXiv:1204.0913] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Walls, Lines and Spectral Dualities in 3d Gauge Theories, JHEP 05 (2014) 047 [arXiv:1302.0015] [INSPIRE].
K. Bulycheva, H.-Y. Chen, A. Gorsky and P. Koroteev, BPS states in omega background and integrability, JHEP 10 (2012) 116 [arXiv:1207.0460] [INSPIRE].
H.-Y. Chen, P.-S. Hsin and P. Koroteev, On the Integrability of Four Dimensional \( \mathcal{N} \) = 2 Gauge Theories in the omega Background, JHEP 08 (2013) 076 [arXiv:1305.5614] [INSPIRE].
Y. Luo, M.-C. Tan and J. Yagi, \( \mathcal{N} \) = 2 supersymmetric gauge theories and quantum integrable systems, JHEP 03 (2014) 090 [arXiv:1310.0827] [INSPIRE].
G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N ) Quantum Intermediate Long Wave Hydrodynamics, JHEP 07 (2014) 141 [arXiv:1403.6454] [INSPIRE].
G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Quantum Cohomology and Quantum Hydrodynamics from Supersymmetric Quiver Gauge Theories, J. Geom. Phys. 109 (2016) 3 [arXiv:1505.07116] [INSPIRE].
P. Koroteev and A. Sciarappa, Quantum Hydrodynamics from Large-N Supersymmetric Gauge Theories, arXiv:1510.00972 [INSPIRE].
J.-E. Bourgine, Y. Matsuo and H. Zhang, Holomorphic field realization of SH c and quantum geometry of quiver gauge theories, JHEP 04 (2016) 167 [arXiv:1512.02492] [INSPIRE].
J. Lamers, The Bethe/Gauge Correspondence, A mysterious link between quantum integrability and supersymmetric gauge theory, MSc Thesis, Utrecht University, Utrecht The Netherlands (2012) http://dspace.library.uu.nl/handle/1874/253835.
N. Dorey, S. Lee and T.J. Hollowood, Quantization of Integrable Systems and a 2d/4d Duality, JHEP 10 (2011) 077 [arXiv:1103.5726] [INSPIRE].
H.-Y. Chen, N. Dorey, T.J. Hollowood and S. Lee, A New 2d/4d Duality via Integrability, JHEP 09 (2011) 040 [arXiv:1104.3021] [INSPIRE].
H.-Y. Chen, T.J. Hollowood and P. Zhao, A 5d/3d duality from relativistic integrable system, JHEP 07 (2012) 139 [arXiv:1205.4230] [INSPIRE].
F. Nieri, S. Pasquetti, F. Passerini and A. Torrielli, 5D partition functions, q-Virasoro systems and integrable spin-chains, JHEP 12 (2014) 040 [arXiv:1312.1294] [INSPIRE].
M. Bullimore, H.-C. Kim and P. Koroteev, Defects and Quantum Seiberg-Witten Geometry, JHEP 05 (2015) 095 [arXiv:1412.6081] [INSPIRE].
D. Orlando and S. Reffert, Relating Gauge Theories via Gauge/Bethe Correspondence, JHEP 10 (2010) 071 [arXiv:1005.4445] [INSPIRE].
F. Benini, D.S. Park and P. Zhao, Cluster Algebras from Dualities of 2d \( \mathcal{N} \) = (2, 2) Quiver Gauge Theories, Commun. Math. Phys. 340 (2015) 47 [arXiv:1406.2699] [INSPIRE].
D. Gaiotto and P. Koroteev, On Three Dimensional Quiver Gauge Theories and Integrability, JHEP 05 (2013) 126 [arXiv:1304.0779] [INSPIRE].
K.K. Kozlowski and J. Teschner, TBA for the Toda chain, in proceedings of the Infinite Analysis 09. New Trends in Quantum Integrable Systems, Kyoto, Japan, 27-31 July 2009, B. Feigin, M. Jimbo and M. Okado eds., World Scientific Publishing Co. Pte. Ltd. (2011), pp. 195-219 [ISBN: 978-981-4324-36-6] [arXiv:1006.2906] [INSPIRE].
C. Meneghelli and G. Yang, Mayer-Cluster Expansion of Instanton Partition Functions and Thermodynamic Bethe Ansatz, JHEP 05 (2014) 112 [arXiv:1312.4537] [INSPIRE].
J.-E. Bourgine, Confinement and Mayer cluster expansions, Int. J. Mod. Phys. A 29 (2014) 1450077 [arXiv:1402.1626] [INSPIRE].
Y. Hatsuda and M. Mariño, Exact quantization conditions for the relativistic Toda lattice, JHEP 05 (2016) 133 [arXiv:1511.02860] [INSPIRE].
N.A. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional \( \mathcal{N} \) = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
N.A. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
V.I. Inozemtsev, The Finite Toda Lattices, Commun. Math. Phys. 121 (1989) 629.
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].
A. Hanany and D. Tong, Vortex strings and four-dimensional gauge dynamics, JHEP 04 (2004) 066 [hep-th/0403158] [INSPIRE].
A. Hanany and K. Hori, Branes and \( \mathcal{N} \) = 2 theories in two-dimensions, Nucl. Phys. B 513 (1998) 119 [hep-th/9707192] [INSPIRE].
N. Dorey, The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms, JHEP 11 (1998) 005 [hep-th/9806056] [INSPIRE].
N. Dorey, T.J. Hollowood and D. Tong, The BPS spectra of gauge theories in two-dimensions and four-dimensions, JHEP 05 (1999) 006 [hep-th/9902134] [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in proceedings of the Conference on the Mathematical Beauty of Physics (In Memory of C. Itzykson), Saclay, France, 5-7 June 1996 [hep-th/9607163] [INSPIRE].
A. Kapustin, Solution of \( \mathcal{N} \) = 2 gauge theories via compactification to three-dimensions, Nucl. Phys. B 534 (1998) 531 [hep-th/9804069] [INSPIRE].
A. Kapustin and S. Sethi, The Higgs branch of impurity theories, Adv. Theor. Math. Phys. 2 (1998) 571 [hep-th/9804027] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
N. Dorey and A. Singleton, Instantons, Integrability and Discrete Light-Cone Quantisation, arXiv:1412.5178 [INSPIRE].
N.A. Nekrasov, Holomorphic bundles and many body systems, Commun. Math. Phys. 180 (1996) 587 [hep-th/9503157] [INSPIRE].
N. Dorey, T.J. Hollowood and S.P. Kumar, An Exact elliptic superpotential for \( \mathcal{N} \) = 1∗ deformations of finite \( \mathcal{N} \) = 2 gauge theories, Nucl. Phys. B 624 (2002) 95 [hep-th/0108221] [INSPIRE].
J. Gibbons and T. Hermsen, A generalisation of the Calogero-Moser system, Physica D 11 (1984) 337.
J.A. Minahan and A.P. Polychronakos, Integrable systems for particles with internal degrees of freedom, Phys. Lett. B 302 (1993) 265 [hep-th/9206046] [INSPIRE].
Z.N.C. Ha and F.D.M. Haldane, Models with inverse-square exchange, Phys. Rev. B 46 (1992) 9359 [INSPIRE].
I. Krichever, O. Babelon, E. Billey and M. Talon, Spin generalization of the Calogero-Moser system and the matrix KP equation, Amer. Math. Soc. Transl. Ser. 2 170 (1995) 92 [hep-th/9411160] [INSPIRE].
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].
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press (1997).
K. Hikami and M. Wadati, Integrability of Calogero-Moser spin system, J. Phys. Soc. Jpn. 62 (1993) 469 [INSPIRE].
M.A. Olshanetsky and A.M. Perelomov, Classical integrable finite dimensional systems related to Lie algebras, Phys. Rept. 71 (1981) 313 [INSPIRE].
P.P. Kulish, N.Y. Reshetikhin and E.K. Sklyanin, Yang-Baxter Equation and Representation Theory. 1., Lett. Math. Phys. 5 (1981) 393 [INSPIRE].
L.D. Faddeev, Algebraic aspects of Bethe Ansatz, Int. J. Mod. Phys. A 10 (1995) 1845 [hep-th/9404013] [INSPIRE].
L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in proceedings of the Les Houches School of Physics: Astrophysical Sources of Gravitational Radiation, Les Houches, France, 26 September-6 October 1995, pp. 149-219 [hep-th/9605187] [INSPIRE].
M. Humi, Separation of coupled systems of differential equations by Darboux transformations, J. Phys. A 18 (1985) 1085.
F. Cannata and M.V. Ioffe, Coupled-channel scattering and separation of coupled differential equations by generalized Darboux transformations, J. Phys. A 26 (1993) L89 [INSPIRE].
S.R. Coleman and H.J. Thun, On the Prosaic Origin of the Double Poles in the sine-Gordon S-Matrix, Commun. Math. Phys. 61 (1978) 31 [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N.A. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
E. Witten, Phases of \( \mathcal{N} \) = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
E. D’Hoker and D.H. Phong, Calogero-Moser and Toda systems for twisted and untwisted affine Lie algebras, Nucl. Phys. B 530 (1998) 611 [hep-th/9804125] [INSPIRE].
S.P. Khastgir, R. Sasaki and K. Takasaki, Calogero-Moser models. 4. Limits to Toda theory, Prog. Theor. Phys. 102 (1999) 749 [hep-th/9907102] [INSPIRE].
Y. Chernyakov and A. Zotov, Integrable many body systems via Inosemtsev limit, Theor. Math. Phys. 129 (2001) 1526 [hep-th/0102069] [INSPIRE].
M. Jimbo, Quantum R matrix for the generalized Toda system, Commun. Math. Phys. 102 (1986) 537 [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: 1512.09367
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
Dorey, N., Zhao, P. Solution of quantum integrable systems from quiver gauge theories. J. High Energ. Phys. 2017, 118 (2017). https://doi.org/10.1007/JHEP02(2017)118
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2017)118