Abstract
We conjecture closed-form expressions for the Macdonald limits of the super-conformal indices of the (A 1 , A 2n − 3) and (A 1 , D 2n ) Argyres-Douglas (AD) theories in terms of certain simple deformations of Macdonald polynomials. As checks of our conjectures, we demonstrate compatibility with two S-dualities, we show symmetry enhancement for special values of n, and we argue that our expressions encode a non-trivial set of renormalization group flows. Moreover, we demonstrate that, for certain values of n, our conjectures imply simple operator relations involving composites built out of the SU(2) R currents and flavor symmetry moment maps, and we find a consistent picture in which these relations give rise to certain null states in the corresponding chiral algebras. In addition, we show that the Hall-Littlewood limits of our indices are equivalent to the corresponding Higgs branch Hilbert series. We explain this fact by considering the S 1 reductions of our theories and showing that the equivalence follows from an inequality on monopole quantum numbers whose coefficients are fixed by data of the four-dimensional parent theories. Finally, we comment on the implications of our work for more general \( \mathcal{N}=2 \) superconformal field theories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
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].
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].
D. Gaiotto, N. Seiberg and Y. Tachikawa, Comments on scaling limits of 4d N = 2 theories, JHEP 01 (2011) 078 [arXiv:1011.4568] [INSPIRE].
M. Buican and T. Nishinaka, On the superconformal index of Argyres-Douglas theories, J. Phys. A 49 (2016) 015401 [arXiv:1505.05884] [INSPIRE].
M. Buican and T. Nishinaka, Argyres-Douglas Theories, S 1 Reductions and Topological Symmetries, J. Phys. A 49 (2016) 045401 [arXiv:1505.06205] [INSPIRE].
C. Cordova and S.-H. Shao, Schur Indices, BPS Particles and Argyres-Douglas Theories, JHEP 01 (2016) 040 [arXiv:1506.00265] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP 02 (2012) 031 [arXiv:1112.1691] [INSPIRE].
D. Xie, General Argyres-Douglas Theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d Superconformal Index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].
Y. Wang and D. Xie, Classification of Argyres-Douglas theories from M5 branes, arXiv:1509.00847 [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
M. Buican, T. Nishinaka and C. Papageorgakis, Constraints on chiral operators in \( \mathcal{N}=2 \) SCFTs, JHEP 12 (2014) 095 [arXiv:1407.2835] [INSPIRE].
F.A. Dolan and H. Osborn, On short and semi-short representations for four-dimensional superconformal symmetry, Annals Phys. 307 (2003) 41 [hep-th/0209056] [INSPIRE].
V.K. Dobrev and V.B. Petkova, All Positive Energy Unitary Irreducible Representations of Extended Conformal Supersymmetry, Phys. Lett. B 162 (1985) 127 [INSPIRE].
M. Buican and T. Nishinaka, On Operator Constraints in Argyres-Douglas Theories.
M. Del Zotto and A. Hanany, Complete Graphs, Hilbert Series and the Higgs branch of the 4d \( \mathcal{N}=2 \) (A n , A m ) SCFTs, Nucl. Phys. B 894 (2015) 439 [arXiv:1403.6523] [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].
M. Buican, S. Giacomelli, T. Nishinaka and C. Papageorgakis, Argyres-Douglas Theories and S-duality, JHEP 02 (2015) 185 [arXiv:1411.6026] [INSPIRE].
M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and \( 6{\mathrm{d}}_{\left(1,0\right)}\to\ 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} \), JHEP 11 (2015) 123 [arXiv:1504.08348] [INSPIRE].
S. Cecotti and M. Del Zotto, Higher S-dualities and Shephard-Todd groups, JHEP 09 (2015) 035 [arXiv:1507.01799] [INSPIRE].
S. Cecotti, Categorical Tinkertoys for N = 2 Gauge Theories, Int. J. Mod. Phys. A 28 (2013) 1330006 [arXiv:1203.6734] [INSPIRE].
D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].
H. Kanno, K. Maruyoshi, S. Shiba and M. Taki, W 3 irregular states and isolated N = 2 superconformal field theories, JHEP 03 (2013) 147 [arXiv:1301.0721] [INSPIRE].
P.C. Argyres, K. Maruyoshi and Y. Tachikawa, Quantum Higgs branches of isolated N = 2 superconformal field theories, JHEP 10 (2012) 054 [arXiv:1206.4700] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
D. Bowman, A general Heine transformation and symmetric polynomials of Rogers, Prog. Math. 138 (1996) 163.
F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].
S.S. Razamat and B. Willett, Down the rabbit hole with theories of class S, JHEP 10 (2014) 99 [arXiv:1403.6107] [INSPIRE].
K. Maruyoshi, C.Y. Park and W. Yan, BPS spectrum of Argyres-Douglas theory via spectral network, JHEP 12 (2013) 092 [arXiv:1309.3050] [INSPIRE].
P. Liendo, I. Ramirez and J. Seo, Stress-tensor OPE in \( \mathcal{N}=2 \) superconformal theories, JHEP 02 (2016) 019 [arXiv:1509.00033] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
Z. Komargodski, The Constraints of Conformal Symmetry on RG Flows, JHEP 07 (2012) 069 [arXiv:1112.4538] [INSPIRE].
K.A. Intriligator, IR free or interacting? A Proposed diagnostic, Nucl. Phys. B 730 (2005) 239 [hep-th/0509085] [INSPIRE].
M. Buican, A Conjectured Bound on Accidental Symmetries, Phys. Rev. D 85 (2012) 025020 [arXiv:1109.3279] [INSPIRE].
M. Buican, Non-Perturbative Constraints on Light Sparticles from Properties of the RG Flow, JHEP 10 (2014) 26 [arXiv:1206.3033] [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: 1509.05402
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
Buican, M., Nishinaka, T. Argyres-Douglas theories, the Macdonald index, and an RG inequality. J. High Energ. Phys. 2016, 159 (2016). https://doi.org/10.1007/JHEP02(2016)159
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2016)159