Abstract
There are two major ways of constructing 4d \( \mathcal{N} \) = 2 superconformal field theories (SCFTs): the first one is putting a 6d (2, 0) theory on a punctured Riemann surface (class-S theory), and the second one is putting type IIB string theory on a 3d canonical singularity. As there are interests on low rank theories, we search all the possibilities from above two constructions. Most of those theories are engineered by class-S theory with irregular singularities, and we find a universal formula for the rank of theory so that a complete search is possible. We then compute various physical quantities of those theories, such as the central charges, flavor symmetry, associated vertex operator algebra and Higgs branch, etc. One of interesting consequence of our results are the prediction of many new isomorphism of 2d vertex operator algebra.
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P.S. Howe, K.S. Stelle and P.C. West, A Class of Finite Four-Dimensional Supersymmetric Field Theories, Phys. Lett. B 124 (1983) 55 [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013) 239 [arXiv:0907.3987] [INSPIRE].
D. Xie, General Argyres-Douglas Theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
Y. Wang and D. Xie, Classification of Argyres-Douglas theories from M5 branes, Phys. Rev. D 94 (2016) 065012 [arXiv:1509.00847] [INSPIRE].
Y. Wang and D. Xie, Codimension-two defects and Argyres-Douglas theories from outer-automorphism twist in 6d (2, 0) theories, Phys. Rev. D 100 (2019) 025001 [arXiv:1805.08839] [INSPIRE].
D. Xie and S.-T. Yau, 4d N = 2 SCFT and singularity theory Part I: Classification, arXiv:1510.01324 [INSPIRE].
B. Chen et al., 4D \( \mathcal{N} \) = 2 SCFT and singularity theory. Part II: complete intersection, Adv. Theor. Math. Phys. 21 (2017) 121 [arXiv:1604.07843] [INSPIRE].
Y. Wang, D. Xie, S.S.T. Yau and S.-T. Yau, 4d \( \mathcal{N} \) = 2 SCFT from complete intersection singularity, Adv. Theor. Math. Phys. 21 (2017) 801 [arXiv:1606.06306] [INSPIRE].
B. Chen et al., 4d \( \mathcal{N} \) = 2 SCFT and singularity theory Part III: Rigid singularity, Adv. Theor. Math. Phys. 22 (2018) 1885 [arXiv:1712.00464] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].
D. Xie, 3d mirror for Argyres-Douglas theories, arXiv:2107.05258 [INSPIRE].
C. Closset, S. Schafer-Nameki and Y.-N. Wang, Coulomb and Higgs Branches from Canonical Singularities: Part 0, JHEP 02 (2021) 003 [arXiv:2007.15600] [INSPIRE].
C. Closset, S. Giacomelli, S. Schafer-Nameki and Y.-N. Wang, 5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities, JHEP 05 (2021) 274 [arXiv:2012.12827] [INSPIRE].
C. Closset, S. Schäfer-Nameki and Y.-N. Wang, Coulomb and Higgs branches from canonical singularities. Part I. Hypersurfaces with smooth Calabi-Yau resolutions, JHEP 04 (2022) 061 [arXiv:2111.13564] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part I: physical constraints on relevant deformations, JHEP 02 (2018) 001 [arXiv:1505.04814] [INSPIRE].
P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part II: construction of special Kähler geometries and RG flows, JHEP 02 (2018) 002 [arXiv:1601.00011] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part III: enhanced Coulomb branches and central charges, JHEP 02 (2018) 003 [arXiv:1609.04404] [INSPIRE].
P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Expanding the landscape of \( \mathcal{N} \) = 2 rank 1 SCFTs, JHEP 05 (2016) 088 [arXiv:1602.02764] [INSPIRE].
P.C. Argyres, C. Long and M. Martone, The Singularity Structure of Scale-Invariant Rank-2 Coulomb Branches, JHEP 05 (2018) 086 [arXiv:1801.01122] [INSPIRE].
M. Caorsi and S. Cecotti, Geometric classification of 4d \( \mathcal{N} \) = 2 SCFTs, JHEP 07 (2018) 138 [arXiv:1801.04542] [INSPIRE].
P.C. Argyres and M. Martone, The rank 2 classification problem I: scale invariant geometries, arXiv:2209.09248 [INSPIRE].
P.C. Argyres and M. Martone, The rank 2 classification problem II: mapping scale-invariant solutions to SCFTs, arXiv:2209.09911 [INSPIRE].
P.C. Argyres and M. Martone, The rank-2 classification problem III: curves with additional automorphisms, arXiv:2209.10555 [INSPIRE].
D. Xie, On rank two theories with eight supercharges part I: local singularities, arXiv:2212.02472 [INSPIRE].
D. Xie, \( \mathcal{N} \) = 2 SCFT with minimal flavor central charge, arXiv:1712.03244 [INSPIRE].
P. Deligne, La série exceptionnelle de groupes de Lie, Comptes Rendus de l’Academie des Sciences-Serie I-Mathematique 322 (1996) 321.
K. Yonekura, Instanton operators and symmetry enhancement in 5d supersymmetric quiver gauge theories, JHEP 07 (2015) 167 [arXiv:1505.04743] [INSPIRE].
S. Giacomelli, M. Martone, Y. Tachikawa and G. Zafrir, More on \( \mathcal{N} \) = 2 S-folds, JHEP 01 (2021) 054 [arXiv:2010.03943] [INSPIRE].
D. Nanopoulos and D. Xie, Hitchin Equation, Singularity, and N = 2 Superconformal Field Theories, JHEP 03 (2010) 043 [arXiv:0911.1990] [INSPIRE].
O. Chacaltana, J. Distler and Y. Tachikawa, Gaiotto duality for the twisted A2N−1 series, JHEP 05 (2015) 075 [arXiv:1212.3952] [INSPIRE].
O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6d N = (2, 0) theories, Int. J. Mod. Phys. A 28 (2013) 1340006 [arXiv:1203.2930] [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for the DN series, JHEP 02 (2013) 110 [arXiv:1106.5410] [INSPIRE].
O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Twisted D-Series, JHEP 04 (2015) 173 [arXiv:1309.2299] [INSPIRE].
O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the E6 theory, JHEP 09 (2015) 007 [arXiv:1403.4604] [INSPIRE].
O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Twisted E6 Theory, arXiv:1501.00357 [INSPIRE].
O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Z3-twisted D4 Theory, arXiv:1601.02077 [INSPIRE].
O. Chacaltana, J. Distler, A. Trimm and Y. Zhu, Tinkertoys for the E7 theory, JHEP 05 (2018) 031 [arXiv:1704.07890] [INSPIRE].
O. Chacaltana, J. Distler, A. Trimm and Y. Zhu, Tinkertoys for the E8 Theory, arXiv:1802.09626 [INSPIRE].
D. Xie and K. Ye, Argyres-Douglas matter and S-duality: Part II, JHEP 03 (2018) 186 [arXiv:1711.06684] [INSPIRE].
P. Shan, D. Xie and W. Yan, Four dimensional simplectic duality, in preparation.
A. Oblomkov and Z. Yun, Geometric representations of graded and rational Cherednik algebras, Adv. Math. 292 (2016) 601.
A.D. Shapere and Y. Tachikawa, Central charges of N = 2 superconformal field theories in four dimensions, JHEP 09 (2008) 109 [arXiv:0804.1957] [INSPIRE].
D. Xie and P. Zhao, Central charges and RG flow of strongly-coupled N = 2 theory, JHEP 03 (2013) 006 [arXiv:1301.0210] [INSPIRE].
C. Beem et al., Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
D. Xie and W. Yan, Schur sector of Argyres-Douglas theory and W-algebra, SciPost Phys. 10 (2021) 080 [arXiv:1904.09094] [INSPIRE].
V. Kac, S.-S. Roan and M. Wakimoto, Quantum Reduction for Affine Superalgebras, Commun. Math. Phys. 241 (2003) 307.
V.G. Kac and M. Wakimoto, On rationality of W-algebras, arXiv:0711.2296.
D. Xie and W. Yan, 4d \( \mathcal{N} \) = 2 SCFTs and lisse W-algebras, JHEP 04 (2021) 271 [arXiv:1910.02281] [INSPIRE].
J. de Boer and T. Tjin, The Relation between quantum W algebras and Lie algebras, Commun. Math. Phys. 160 (1994) 317 [hep-th/9302006] [INSPIRE].
A.G. Elashvili and V.G. Kac, Classification of good gradings of simple Lie algebras, Am. Math. Soc. Transl. Series 2 213 (2005) 85.
J. Brundan and S.M. Goodwin, Good grading polytopes, Proc. Lond. Math. Soc. 94 (2007) 155.
T. Arakawa, J. Van Ekeren and A. Moreau, Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras, arXiv:2102.13462 [INSPIRE].
T. Arakawa, Chiral algebras of class \( \mathcal{S} \) and Moore-Tachikawa symplectic varieties, arXiv:1811.01577 [INSPIRE].
D. Adamović et al., Conformal embeddings of affine vertex algebras in minimal W-algebras I: structural results, J. Algebra 500 (2018) 117 [arXiv:1602.04687] [INSPIRE].
D. Adamović et al., An Application of Collapsing Levels to the Representation Theory of Affine Vertex Algebras, Int. Math. Res. Not. 2020 (2020) 4103 [INSPIRE].
D. Adamovic, P.M. Frajria and P. Papi, New approaches for studying conformal embeddings and collapsing levels for W-algebras, arXiv:2203.08497 [INSPIRE].
J. Kaidi and M. Martone, New rank-2 Argyres-Douglas theory, Phys. Rev. D 104 (2021) 085004 [arXiv:2104.13929] [INSPIRE].
D. Collingwood and W. McGovern, Argyres?Douglas theories, S1 reductions, and topological symmetries, J. Phys. A49 (2016) 045401 [arXiv:1505.06205].
P.C. Argyres and M. Martone, 4d \( \mathcal{N} \) = 2 theories with disconnected gauge groups, JHEP 03 (2017) 145 [arXiv:1611.08602] [INSPIRE].
D. Xie, W. Yan and S.-T. Yau, Chiral algebra of the Argyres-Douglas theory from M5 branes, Phys. Rev. D 103 (2021) 065003 [arXiv:1604.02155] [INSPIRE].
Acknowledgments
BL, DX and WY are supported by Yau Mathematical Science Center at Tsinghua University. DX and WY are supported by national key research and development program of China (NO. 2020YFA0713000), and NNSF of China with Grant NO: 11847301 and 12047502. BL and WY are supported by Dushi program of Tsinghua.
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Li, B., Xie, D. & Yan, W. On low rank 4d \( \mathcal{N} \) = 2 SCFTs. J. High Energ. Phys. 2023, 132 (2023). https://doi.org/10.1007/JHEP05(2023)132
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DOI: https://doi.org/10.1007/JHEP05(2023)132