Abstract
We study the quantum spectral curve of the Argyres-Douglas theories in the Nekrasov-Sahashvili limit of the Omega-background. Using the ODE/IM correspondence we investigate the quantum integrable model corresponding to the quantum spectral curve. We show that the models for the A 2N -type theories are non-unitary coset models (A 1)1 × (A 1) L /(A 1) L+1 at the fractional level \( L=\frac{2}{2N+1}-2 \), which appear in the study of the 4d/2d correspondence of \( \mathcal{N} \) = 2 superconformal field theories. Based on the WKB analysis, we clarify the relation between the Y-functions and the quantum periods and study the exact Bohr-Sommerfeld quantization condition for the quantum periods. We also discuss the quantum spectral curves for the D and E type theories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
T. Eguchi, K. Hori, K. Ito and S.-K. Yang, Study of N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 471 (1996) 430 [hep-th/9603002] [INSPIRE].
Y. Wang and D. Xie, Classification of Argyres-Douglas theories from M 5 branes, Phys. Rev. D 94 (2016) 065012 [arXiv:1509.00847] [INSPIRE].
D. Xie, General Argyres-Douglas theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Cordova and S.-H. Shao, Schur indices, BPS particles and Argyres-Douglas theories, JHEP 01 (2016) 040 [arXiv:1506.00265] [INSPIRE].
M. Buican and T. Nishinaka, On the superconformal index of Argyres-Douglas theories, J. Phys. A 49 (2016) 015401 [arXiv:1505.05884] [INSPIRE].
D. Xie, W. Yan and S.-T. Yau, Chiral algebra of Argyres-Douglas theory from M5 brane, arXiv:1604.02155 [INSPIRE].
L. Fredrickson, D. Pei, W. Yan and K. Ye, Argyres-Douglas theories, chiral algebras and wild Hitchin characters, arXiv:1701.08782 [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE].
A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].
D. Gaiotto, Opers and TBA, arXiv:1403.6137 [INSPIRE].
S. Cecotti and M. Del Zotto, Y systems, Q systems and 4D N = 2 supersymmetric QFT, J. Phys. A 47 (2014) 474001 [arXiv:1403.7613] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].
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].
P. Dorey, C. Dunning and R. Tateo, The ODE/IM correspondence, J. Phys. A 40 (2007) R205 [hep-th/0703066] [INSPIRE].
P. Dorey, C. Dunning, D. Masoero, J. Suzuki and R. Tateo, Pseudo-differential equations and the Bethe ansatz for the classical Lie algebras, Nucl. Phys. B 772 (2007) 249 [hep-th/0612298] [INSPIRE].
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, 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].
K. Ito and C. Locke, ODE/IM correspondence and modified affine Toda field equations, Nucl. Phys. B 885 (2014) 600 [arXiv:1312.6759] [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].
J. Sun, Polynomial relations for q-characters via the ODE/IM correspondence, SIGMA 8 (2012) 028 [arXiv:1201.1614] [INSPIRE].
A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391 [INSPIRE].
A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen, Simple singularities and N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 344 (1995) 169 [hep-th/9411048] [INSPIRE].
E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].
K. Ito, A-D-E singularity and the Seiberg-Witten theory, Prog. Theor. Phys. Suppl. 135 (1999) 94 [hep-th/9906023] [INSPIRE].
P. Dorey, C. Dunning and R. Tateo, Differential equations for general SU(N ) Bethe ansatz systems, J. Phys. A 33 (2000) 8427 [hep-th/0008039] [INSPIRE].
Y. Sibuya, Global theory of a second-order linear ordinary differential operator with polynomial coefficient, North-Holland, Amsterdam The Netherlands, (1975).
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].
P. Mathieu and M.A. Walton, Fractional level Kac-Moody algebras and nonunitarity coset conformal theories, Prog. Theor. Phys. Suppl. 102 (1990) 229 [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].
L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y -system for scattering amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable quantum field theories in finite volume: excited state energies, Nucl. Phys. B 489 (1997) 487 [hep-th/9607099] [INSPIRE].
F. Ravanini, R. Tateo and A. Valleriani, Dynkin TBAs, Int. J. Mod. Phys. A 8 (1993) 1707 [hep-th/9207040] [INSPIRE].
P. Dorey, C. Dunning, F. Gliozzi and R. Tateo, On the ODE/IM correspondence for minimal models, J. Phys. A 41 (2008) 132001 [arXiv:0712.2010] [INSPIRE].
C. Dunning, Massless flows between minimal W models, Phys. Lett. B 537 (2002) 297 [hep-th/0204090] [INSPIRE].
P.B. Gilkey and G.M. Seitz, Some representations of exceptional Lie algebras, Geom. Dedicata 25 (1988) 407.
N.A. Vavilov, Do it yourself: the structure constants for Lie algebras of types E l , J. Math. Sci. 120 (2004) 1513.
B. Feigin and E. Frenkel, Quantization of soliton systems and Langlands duality, arXiv:0705.2486 [INSPIRE].
T. Creutzig, W -algebras for Argyres-Douglas theories, arXiv:1701.05926 [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].
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: 1707.03596
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
Ito, K., Shu, H. ODE/IM correspondence and the Argyres-Douglas theory. J. High Energ. Phys. 2017, 71 (2017). https://doi.org/10.1007/JHEP08(2017)071
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2017)071