Abstract
We study irregular states of rank-two and three in Liouville theory, based on an ansatz proposed by D. Gaiotto and J. Teschner. Using these irregular states, we evaluate asymptotic expansions of irregular conformal blocks corresponding to the partition functions of (A1, A3) and (A1, D4) Argyres-Douglas theories for general Ω-background parameters. In the limit of vanishing Liouville charge, our result reproduces strong coupling expansions of the partition functions recently obtained via the Painlevé/gauge correspondence. This suggests that the irregular conformal block for one irregular singularity of rank 3 on sphere is also related to Painlevé II. We also find that our partition functions are invariant under the action of the Weyl group of flavor symmetries once four and two-dimensional parameters are correctly identified. We finally propose a generalization of this parameter identification to general irregular states of integer rank.
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 \( \mathcal{N} \) = 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 \( \mathcal{N} \) = 2 superconformal field theories in four-dimensions, Nucl. Phys.B 471 (1996) 430 [hep-th/9603002] [INSPIRE].
K. Maruyoshi and J. Song, Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index, Phys. Rev. Lett.118 (2017) 151602 [arXiv:1606.05632] [INSPIRE].
K. Maruyoshi and J. Song, \( \mathcal{N} \) = 1 deformations and RG flows of \( \mathcal{N} \) = 2 SCFTs, JHEP02 (2017) 075 [arXiv:1607.04281] [INSPIRE].
P. Agarwal, K. Maruyoshi and J. Song, \( \mathcal{N} \) = 1 Deformations and RG flows of \( \mathcal{N} \) = 2 SCFTs, part II: non-principal deformations, JHEP12 (2016) 103 [arXiv:1610.05311] [INSPIRE].
K. Maruyoshi, E. Nardoni and J. Song, Landscape of Simple Superconformal Field Theories in 4d, Phys. Rev. Lett.122 (2019) 121601 [arXiv:1806.08353] [INSPIRE].
P. Agarwal, A. Sciarappa and J. Song, \( \mathcal{N} \) = 1 Lagrangians for generalized Argyres-Douglas theories, JHEP10 (2017) 211 [arXiv:1707.04751] [INSPIRE].
S. Benvenuti and S. Giacomelli, Abelianization and sequential confinement in 2 + 1 dimensions, JHEP10 (2017) 173 [arXiv:1706.04949] [INSPIRE].
S. Benvenuti and S. Giacomelli, Lagrangians for generalized Argyres-Douglas theories, JHEP10 (2017) 106 [arXiv:1707.05113] [INSPIRE].
S. Giacomelli, RG flows with supersymmetry enhancement and geometric engineering, JHEP06 (2018) 156 [arXiv:1710.06469] [INSPIRE].
S. Giacomelli, Infrared enhancement of supersymmetry in four dimensions, JHEP10 (2018) 041 [arXiv:1808.00592] [INSPIRE].
F. Carta, S. Giacomelli and R. Savelli, SUSY enhancement from T-branes, JHEP12 (2018) 127 [arXiv:1809.04906] [INSPIRE].
M.-x. Huang and A. Klemm, Holomorphicity and Modularity in Seiberg-Witten Theories with Matter, JHEP07 (2010) 083 [arXiv:0902.1325] [INSPIRE].
G. Bonelli, O. Lisovyy, K. Maruyoshi, A. Sciarappa and A. Tanzini, On Painlevé/gauge theory correspondence, arXiv:1612.06235 [INSPIRE].
O. Gamayun, N. Iorgov and O. Lisovyy, Conformal field theory of Painlevé VI, JHEP10 (2012) 038 [Erratum ibid.10 (2012) 183] [arXiv:1207.0787] [INSPIRE].
O. Gamayun, N. Iorgov and O. Lisovyy, How instanton combinatorics solves Painlevé VI, V and IIIs, J. Phys.A 46 (2013) 335203 [arXiv:1302.1832] [INSPIRE].
N. Iorgov, O. Lisovyy and J. Teschner, Isomonodromic tau-functions from Liouville conformal blocks, Commun. Math. Phys.336 (2015) 671 [arXiv:1401.6104] [INSPIRE].
M.A. Bershtein and A.I. Shchechkin, Bilinear equations on Painlevé τ functions from CFT, Commun. Math. Phys.339 (2015) 1021 [arXiv:1406.3008] [INSPIRE].
P. Gavrylenko and O. Lisovyy, Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions, Commun. Math. Phys.363 (2018) 1 [arXiv:1608.00958] [INSPIRE].
K. Sakai, BPS index and 4d \( \mathcal{N} \) = 2 superconformal field theories, JHEP07 (2016) 046 [arXiv:1603.09108] [INSPIRE].
K. Ito and T. Okubo, Quantum periods for \( \mathcal{N} \) = 2 SU(2) SQCD around the superconformal point, Nucl. Phys.B 934 (2018) 356 [arXiv:1804.04815] [INSPIRE].
K. Ito, S. Koizumi and T. Okubo, Quantum Seiberg-Witten curve and Universality in Argyres-Douglas theories, Phys. Lett.B 792 (2019) 29 [arXiv:1903.00168] [INSPIRE].
H. Nagoya, Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations, J. Math. Phys.56 (2015) 123505 [arXiv:1505.02398] [INSPIRE].
H. Nagoya, Conformal blocks and Painlevé functions, arXiv:1611.08971 [INSPIRE].
H. Nagoya, Remarks on irregular conformal blocks and Painlevé III and II tau functions, arXiv:1804.04782 [INSPIRE].
O. Lisovyy, H. Nagoya and J. Roussillon, Irregular conformal blocks and connection formulae for Painlevé V functions, J. Math. Phys.59 (2018) 091409 [arXiv:1806.08344] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys.91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
D. Gaiotto, Asymptotically free \( \mathcal{N} \) = 2 theories and irregular conformal blocks, J. Phys. Conf. Ser.462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP02 (2012) 031 [arXiv:1112.1691] [INSPIRE].
D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I, JHEP12 (2012) 050 [arXiv:1203.1052] [INSPIRE].
H. Kanno, K. Maruyoshi, S. Shiba and M. Taki, W 3irregular states and isolated \( \mathcal{N} \) = 2 superconformal field theories, JHEP03 (2013) 147 [arXiv:1301.0721] [INSPIRE].
Y. Matsuo, C. Rim and H. Zhang, Construction of Gaiotto states with fundamental multiplets through Degenerate DAHA, JHEP09 (2014) 028 [arXiv:1405.3141] [INSPIRE].
D. Polyakov and C. Rim, Irregular Vertex Operators for Irregular Conformal Blocks, Phys. Rev.D 93 (2016) 106002 [arXiv:1601.07756] [INSPIRE].
D. Polyakov and C. Rim, Vertex Operators for Irregular Conformal Blocks: Supersymmetric Case, Phys. Rev.D 94 (2016) 086011 [arXiv:1604.08741] [INSPIRE].
C. Rim and H. Zhang, Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation, Nucl. Phys.B 919 (2017) 182 [arXiv:1608.05027] [INSPIRE].
A. Grassi and J. Gu, Argyres-Douglas theories, Painlevé II and quantum mechanics, JHEP02 (2019) 060 [arXiv:1803.02320] [INSPIRE].
H. Itoyama, T. Oota and K. Yano, Discrete Painleve system and the double scaling limit of the matrix model for irregular conformal block and gauge theory, Phys. Lett.B 789 (2019) 605 [arXiv:1805.05057] [INSPIRE].
H. Itoyama, T. Oota and K. Yano, Discrete Painleve system for the partition function of N f = 2 SU(2) supersymmetric gauge theory and its double scaling limit, arXiv:1812.00811 [INSPIRE].
H. Itoyama, K. Maruyoshi and T. Oota, The Quiver Matrix Model and 2d-4d Conformal Connection, Prog. Theor. Phys.123 (2010) 957 [arXiv:0911.4244] [INSPIRE].
T. Eguchi and K. Maruyoshi, Penner Type Matrix Model and Seiberg-Witten Theory, JHEP02 (2010) 022 [arXiv:0911.4797] [INSPIRE].
R. Schiappa and N. Wyllard, An A Rthreesome: Matrix models, 2d CFTs and 4d \( \mathcal{N} \) = 2 gauge theories, J. Math. Phys.51 (2010) 082304 [arXiv:0911.5337] [INSPIRE].
A. Mironov, A. Morozov and S. Shakirov, Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions, JHEP02 (2010) 030 [arXiv:0911.5721] [INSPIRE].
M. Fujita, Y. Hatsuda and T.-S. Tai, Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model, JHEP03 (2010) 046 [arXiv:0912.2988] [INSPIRE].
H. Itoyama and T. Oota, Method of Generating q-Expansion Coefficients for Conformal Block and \( \mathcal{N} \) = 2 Nekrasov Function by beta-Deformed Matrix Model, Nucl. Phys.B 838 (2010) 298 [arXiv:1003.2929] [INSPIRE].
A. Mironov, A. Morozov and A. Morozov, Conformal blocks and generalized Selberg integrals, Nucl. Phys.B 843 (2011) 534 [arXiv:1003.5752] [INSPIRE].
A. Morozov and S. Shakirov, The matrix model version of AGT conjecture and CIV-DV prepotential, JHEP08 (2010) 066 [arXiv:1004.2917] [INSPIRE].
T. Eguchi and K. Maruyoshi, Seiberg-Witten theory, matrix model and AGT relation, JHEP07 (2010) 081 [arXiv:1006.0828] [INSPIRE].
H. Itoyama, T. Oota and N. Yonezawa, Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type, Phys. Rev.D 82 (2010) 085031 [arXiv:1008.1861] [INSPIRE].
K. Maruyoshi and F. Yagi, Seiberg-Witten curve via generalized matrix model, JHEP01 (2011) 042 [arXiv:1009.5553] [INSPIRE].
G. Bonelli, K. Maruyoshi, A. Tanzini and F. Yagi, Generalized matrix models and AGT correspondence at all genera, JHEP07 (2011) 055 [arXiv:1011.5417] [INSPIRE].
H. Itoyama and N. Yonezawa, ϵ-Corrected Seiberg-Witten Prepotential Obtained From Half Genus Expansion in beta-Deformed Matrix Model, Int. J. Mod. Phys.A 26 (2011) 3439 [arXiv:1104.2738] [INSPIRE].
T. Nishinaka and C. Rim, β-Deformed Matrix Model and Nekrasov Partition Function, JHEP02 (2012) 114 [arXiv:1112.3545] [INSPIRE].
D. Galakhov, A. Mironov and A. Morozov, S-duality as a beta-deformed Fourier transform, JHEP08 (2012) 067 [arXiv:1205.4998] [INSPIRE].
J.-E. Bourgine, Large N limit of beta-ensembles and deformed Seiberg-Witten relations, JHEP08 (2012) 046 [arXiv:1206.1696] [INSPIRE].
T. Nishinaka and C. Rim, Matrix models for irregular conformal blocks and Argyres-Douglas theories, JHEP10 (2012) 138 [arXiv:1207.4480] [INSPIRE].
C. Rim, Irregular conformal block and its matrix model, arXiv:1210.7925 [INSPIRE].
S.-K. Choi and C. Rim, Parametric dependence of irregular conformal block, JHEP04 (2014) 106 [arXiv:1312.5535] [INSPIRE].
H. Itoyama, T. Oota and R. Yoshioka, 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space, Nucl. Phys.B 877 (2013) 506 [arXiv:1308.2068] [INSPIRE].
H. Itoyama, T. Oota and R. Yoshioka, q-Virasoro/W Algebra at Root of Unity and Parafermions, Nucl. Phys.B 889 (2014) 25 [arXiv:1408.4216] [INSPIRE].
S.K. Choi, C. Rim and H. Zhang, Virasoro irregular conformal block and beta deformed random matrix model, Phys. Lett.B 742 (2015) 50 [arXiv:1411.4453] [INSPIRE].
C. Rim and H. Zhang, Classical Virasoro irregular conformal block, JHEP07 (2015) 163 [arXiv:1504.07910] [INSPIRE].
S.K. Choi and C. Rim, Irregular matrix model with \( \mathcal{W} \)symmetry, J. Phys.A 49 (2016) 075201 [arXiv:1506.02421] [INSPIRE].
C. Rim and H. Zhang, Classical Virasoro irregular conformal block II, JHEP09 (2015) 097 [arXiv:1506.03561] [INSPIRE].
S.K. Choi, C. Rim and H. Zhang, Irregular conformal block, spectral curve and flow equations, JHEP03 (2016) 118 [arXiv:1510.09060] [INSPIRE].
D. Polyakov and C. Rim, Super-spectral curve of irregular conformal blocks, JHEP12 (2016) 004 [arXiv:1608.04921] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th/0206161] [INSPIRE].
D. Xie, General Argyres-Douglas Theory, JHEP01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
S. Cecotti, A. Neitzke and C. Vafa, R-Twisting and 4d/2d Correspondences, arXiv:1006.3435 [INSPIRE].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys.B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
D. Gaiotto, \( \mathcal{N} \) = 2 dualities, JHEP08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
K. Ito, S. Kanno and T. Okubo, Quantum periods and prepotential in \( \mathcal{N} \) = 2 SU(2) SQCD, JHEP08 (2017) 065 [arXiv:1705.09120] [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: 1905.03795
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Nishinaka, T., Uetoko, T. Argyres-Douglas theories and Liouville irregular states. J. High Energ. Phys. 2019, 104 (2019). https://doi.org/10.1007/JHEP09(2019)104
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2019)104