Abstract
We apply the numerical conformal bootstrap to correlators of Coulomb and Higgs branch operators in 4d \( \mathcal{N} \) = 2 superconformal theories. We start by revisiting previous results on single correlators of Coulomb branch operators. In particular, we present improved bounds on OPE coefficients for some selected Argyres-Douglas models, and compare them to recent work where the same cofficients were obtained in the limit of large r charge. There is solid agreement between all the approaches. The improved bounds can be used to extract an approximate spectrum of the Argyres-Douglas models, which can then be used as a guide in order to corner these theories to numerical islands in the space of conformal dimensions. When there is a flavor symmetry present, we complement the analysis by including mixed correlators of Coulomb branch operators and the moment map, a Higgs branch operator which sits in the same multiplet as the flavor current. After calculating the relevant superconformal blocks we apply the numerical machinery to the mixed system. We put general constraints on CFT data appearing in the new channels, with particular emphasis on the simplest Argyres-Douglas model with non-trivial flavor symmetry.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of 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 N = 2 SCFTs. Part II: construction of special Kähler geometries and RG flows, JHEP 02 (2018) 002 [arXiv:1601.00011] [INSPIRE].
P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Expanding the landscape of N = 2 rank 1 SCFTs, JHEP 05 (2016) 088 [arXiv:1602.02764] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of N = 2 SCFTs. Part III: enhanced Coulomb branches and central charges, JHEP 02 (2018) 003 [arXiv:1609.04404] [INSPIRE].
P. Argyres and M. Martone, Construction and classification of Coulomb branch geometries, arXiv:2003.04954 [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. Beem and L. Rastelli, Vertex operator algebras, Higgs branches, and modular differential equations, JHEP 08 (2018) 114 [arXiv:1707.07679] [INSPIRE].
F. Bonetti, C. Meneghelli and L. Rastelli, VOAs labelled by complex reflection groups and 4d SCFTs, JHEP 05 (2019) 155 [arXiv:1810.03612] [INSPIRE].
C. Beem, C. Meneghelli and L. Rastelli, Free field realizations from the Higgs branch, JHEP 09 (2019) 058 [arXiv:1903.07624] [INSPIRE].
C. Beem, C. Meneghelli, W. Peelaers and L. Rastelli, VOAs and rank-two instanton SCFTs, Commun. Math. Phys. 377 (2020) 2553 [arXiv:1907.08629] [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The N = 2 superconformal bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE].
M. Lemos and P. Liendo, Bootstrapping N = 2 chiral correlators, JHEP 01 (2016) 025 [arXiv:1510.03866] [INSPIRE].
M. Cornagliotto, M. Lemos and P. Liendo, Bootstrapping the (A1, A2) Argyres-Douglas theory, JHEP 03 (2018) 033 [arXiv:1711.00016] [INSPIRE].
S. Hellerman, S. Maeda, D. Orlando, S. Reffert and M. Watanabe, Universal correlation functions in rank 1 SCFTs, JHEP 12 (2019) 047 [arXiv:1804.01535] [INSPIRE].
A. Grassi, Z. Komargodski and L. Tizzano, Extremal correlators and random matrix theory, arXiv:1908.10306 [INSPIRE].
S. El-Showk and M.F. Paulos, Bootstrapping conformal field theories with the extremal functional method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].
Z. Li and N. Su, 3D CFT archipelago from single correlator bootstrap, Phys. Lett. B 797 (2019) 134920 [arXiv:1706.06960] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of superconformal symmetry in diverse dimensions, JHEP 03 (2019) 163 [arXiv:1612.00809] [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].
M. Baggio, V. Niarchos and K. Papadodimas, tt* equations, localization and exact chiral rings in 4d N = 2 SCFTs, JHEP 02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, Exact correlation functions in SU(2) N = 2 superconformal QCD, Phys. Rev. Lett. 113 (2014) 251601 [arXiv:1409.4217] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, On exact correlation functions in SU(N) N = 2 superconformal QCD, JHEP 11 (2015) 198 [arXiv:1508.03077] [INSPIRE].
M. Baggio, V. Niarchos, K. Papadodimas and G. Vos, Large-N correlation functions in N = 2 superconformal QCD, JHEP 01 (2017) 101 [arXiv:1610.07612] [INSPIRE].
E. Gerchkovitz, J. Gomis, N. Ishtiaque, A. Karasik, Z. Komargodski and S.S. Pufu, Correlation functions of Coulomb branch operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].
S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT operator spectrum at large global charge, JHEP 12 (2015) 071 [arXiv:1505.01537] [INSPIRE].
S. Hellerman and S. Maeda, On the large R-charge expansion in N = 2 superconformal field theories, JHEP 12 (2017) 135 [arXiv:1710.07336] [INSPIRE].
M. Beccaria, On the large R-charge N = 2 chiral correlators and the Toda equation, JHEP 02 (2019) 009 [arXiv:1809.06280] [INSPIRE].
A. Bourget, D. Rodriguez-Gomez and J.G. Russo, A limit for large R-charge correlators in N = 2 theories, JHEP 05 (2018) 074 [arXiv:1803.00580] [INSPIRE].
P. Liendo, I. Ramirez and J. Seo, Stress-tensor OPE in N = 2 superconformal theories, JHEP 02 (2016) 019 [arXiv:1509.00033] [INSPIRE].
M. Lemos and P. Liendo, N = 2 central charge bounds from 2d chiral algebras, JHEP 04 (2016) 004 [arXiv:1511.07449] [INSPIRE].
C. Beem, Flavor symmetries and unitarity bounds in N = 2 superconformal field theories, Phys. Rev. Lett. 122 (2019) 241603 [arXiv:1812.06099] [INSPIRE].
P.C. Argyres, M. 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].
K. Maruyoshi and J. Song, N = 1 deformations and RG flows of N = 2 SCFTs, JHEP 02 (2017) 075 [arXiv:1607.04281] [INSPIRE].
O. Aharony and Y. Tachikawa, A holographic computation of the central charges of d = 4, N = 2 SCFTs, JHEP 01 (2008) 037 [arXiv:0711.4532] [INSPIRE].
M. Buican and T. Nishinaka, On the superconformal index of Argyres-Douglas theories, J. Phys. A 49 (2016) 015401 [arXiv:1505.05884] [INSPIRE].
C. Cordova and S.-H. Shao, Schur indices, BPS particles, and Argyres-Douglas theories, JHEP 01 (2016) 040 [arXiv:1506.00265] [INSPIRE].
O. Aharony, A. Fayyazuddin and J.M. Maldacena, The large N limit of N = 2, N = 1 field theories from three-branes in F-theory, JHEP 07 (1998) 013 [hep-th/9806159] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
D. Poland and D. Simmons-Duffin, Bounds on 4D conformal and superconformal field theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
A. Fitzpatrick, J. Kaplan, Z.U. Khandker, D. Li, D. Poland and D. Simmons-Duffin, Covariant approaches to superconformal blocks, JHEP 08 (2014) 129 [arXiv:1402.1167] [INSPIRE].
F.A. Dolan and H. Osborn, Superconformal symmetry, correlation functions and the operator product expansion, Nucl. Phys. B 629 (2002) 3 [hep-th/0112251] [INSPIRE].
F.A. Dolan, L. Gallot and E. Sokatchev, On four-point functions of 1/2-BPS operators in general dimensions, JHEP 09 (2004) 056 [hep-th/0405180] [INSPIRE].
M. Nirschl and H. Osborn, Superconformal Ward identities and their solution, Nucl. Phys. B 711 (2005) 409 [hep-th/0407060] [INSPIRE].
G. Arutyunov, B. Eden and E. Sokatchev, On nonrenormalization and OPE in superconformal field theories, Nucl. Phys. B 619 (2001) 359 [hep-th/0105254] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
D. Poland, D. Simmons-Duffin and A. Vichi, Carving out the space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising model with the conformal bootstrap II. c-minimization and precise critical exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N) archipelago, JHEP 11 (2015) 106 [arXiv:1504.07997] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision islands in the Ising and O(N) models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
S.M. Chester et al., Carving out OPE space and precise O(2) model critical exponents, JHEP 06 (2020) 142 [arXiv:1912.03324] [INSPIRE].
A. Atanasov, A. Hillman and D. Poland, Bootstrapping the minimal 3D SCFT, JHEP 11 (2018) 140 [arXiv:1807.05702] [INSPIRE].
J. Rong and N. Su, Bootstrapping minimal N = 1 superconformal field theory in three dimensions, arXiv:1807.04434 [INSPIRE].
J. Rong and N. Su, Bootstrapping the N = 1 Wess-Zumino models in three dimensions, arXiv:1910.08578 [INSPIRE].
N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping SCFTs with four supercharges, JHEP 08 (2015) 142 [arXiv:1503.02081] [INSPIRE].
S.M. Chester, L.V. Iliesiu, S.S. Pufu and R. Yacoby, Bootstrapping O(N) vector models with four supercharges in 3 ≤ d ≤ 4, JHEP 05 (2016) 103 [arXiv:1511.07552] [INSPIRE].
S.M. Chester, S. Giombi, L.V. Iliesiu, I.R. Klebanov, S.S. Pufu and R. Yacoby, Accidental symmetries and the conformal bootstrap, JHEP 01 (2016) 110 [arXiv:1507.04424] [INSPIRE].
J.-B. Bae, D. Gang and J. Lee, 3d N = 2 minimal SCFTs from wrapped M5-branes, JHEP 08 (2017) 118 [arXiv:1610.09259] [INSPIRE].
M. Baggio, N. Bobev, S.M. Chester, E. Lauria and S.S. Pufu, Decoding a three-dimensional conformal manifold, JHEP 02 (2018) 062 [arXiv:1712.02698] [INSPIRE].
S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, The N = 8 superconformal bootstrap in three dimensions, JHEP 09 (2014) 143 [arXiv:1406.4814] [INSPIRE].
S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, Exact correlators of BPS operators from the 3d superconformal bootstrap, JHEP 03 (2015) 130 [arXiv:1412.0334] [INSPIRE].
N.B. Agmon, S.M. Chester and S.S. Pufu, Solving M-theory with the conformal bootstrap, JHEP 06 (2018) 159 [arXiv:1711.07343] [INSPIRE].
N.B. Agmon, S.M. Chester and S.S. Pufu, The M-theory archipelago, JHEP 02 (2020) 010 [arXiv:1907.13222] [INSPIRE].
M. Berkooz, R. Yacoby and A. Zait, Bounds on N = 1 superconformal theories with global symmetries, JHEP 08 (2014) 008 [Erratum ibid. 01 (2015) 132] [arXiv:1402.6068] [INSPIRE].
D. Poland and A. Stergiou, Exploring the minimal 4D N = 1 SCFT, JHEP 12 (2015) 121 [arXiv:1509.06368] [INSPIRE].
D. Li, D. Meltzer and A. Stergiou, Bootstrapping mixed correlators in 4D N = 1 SCFTs, JHEP 07 (2017) 029 [arXiv:1702.00404] [INSPIRE].
M. Lemos, P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping N = 3 superconformal theories, JHEP 04 (2017) 032 [arXiv:1612.01536] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, The N = 4 superconformal bootstrap, Phys. Rev. Lett. 111 (2013) 071601 [arXiv:1304.1803] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, More N = 4 superconformal bootstrap, Phys. Rev. D 96 (2017) 046014 [arXiv:1612.02363] [INSPIRE].
Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, N = 4 superconformal bootstrap of the K 3 CFT, JHEP 05 (2017) 126 [arXiv:1511.04065] [INSPIRE].
Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, (2, 2) superconformal bootstrap in two dimensions, JHEP 05 (2017) 112 [arXiv:1610.05371] [INSPIRE].
M. Cornagliotto, M. Lemos and V. Schomerus, Long multiplet bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [INSPIRE].
C.-M. Chang, M. Fluder, Y.-H. Lin and Y. Wang, Spheres, charges, instantons and bootstrap: a five-dimensional odyssey, JHEP 03 (2018) 123 [arXiv:1710.08418] [INSPIRE].
C. Beem, M. Lemos, L. Rastelli and B.C. van Rees, The (2, 0) superconformal bootstrap, Phys. Rev. D 93 (2016) 025016 [arXiv:1507.05637] [INSPIRE].
C.-M. Chang and Y.-H. Lin, Carving out the end of the world or (superconformal bootstrap in six dimensions), JHEP 08 (2017) 128 [arXiv:1705.05392] [INSPIRE].
P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping the half-BPS line defect, JHEP 10 (2018) 077 [arXiv:1806.01862] [INSPIRE].
A. Gimenez-Grau and P. Liendo, Bootstrapping line defects in N = 2 theories, JHEP 03 (2020) 121 [arXiv:1907.04345] [INSPIRE].
C.-M. Chang, M. Fluder, Y.-H. Lin, S.-H. Shao and Y. Wang, 3d N = 4 bootstrap and mirror symmetry, arXiv:1910.03600 [INSPIRE].
S.M. Chester, Weizmann lectures on the numerical conformal bootstrap, arXiv:1907.05147 [INSPIRE].
D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
J. Song, Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT, JHEP 02 (2016) 045 [arXiv:1509.06730] [INSPIRE].
N. Su, Search methods for the numerical bootstrap, in Developments in the numerical bootstrap, http://scgp.stonybrook.edu/video_portal/video.php?id=4327, November 2019.
I. Buric, V. Schomerus and E. Sobko, Superconformal blocks: general theory, JHEP 01 (2020) 159 [arXiv:1904.04852] [INSPIRE].
I. Burić, V. Schomerus and E. Sobko, The superconformal X -ing equation, JHEP 10 (2020) 147 [arXiv:2005.13547] [INSPIRE].
R. Doobary and P. Heslop, Superconformal partial waves in Grassmannian field theories, JHEP 12 (2015) 159 [arXiv:1508.03611] [INSPIRE].
Z. Li, Superconformal partial waves for stress-tensor multiplet correlator in 4D N = 2 SCFTs, JHEP 05 (2020) 101 [arXiv:1806.11550] [INSPIRE].
Z.U. Khandker, D. Li, D. Poland and D. Simmons-Duffin, N = 1 superconformal blocks for general scalar operators, JHEP 08 (2014) 049 [arXiv:1404.5300] [INSPIRE].
J.-F. Fortin, K. Intriligator and A. Stergiou, Current OPEs in superconformal theories, JHEP 09 (2011) 071 [arXiv:1107.1721] [INSPIRE].
I.A. Ramírez, Mixed OPEs in N = 2 superconformal theories, JHEP 05 (2016) 043 [arXiv:1602.07269] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
D. Karateev, P. Kravchuk, M. Serone and A. Vichi, Fermion conformal bootstrap in 4d, JHEP 06 (2019) 088 [arXiv:1902.05969] [INSPIRE].
P. Kravchuk, W. Landry and D. Simmons-Duffin, Scalar blocks, https://gitlab.com/bootstrapcollaboration/scalar_blocks, (2020).
W. Landry and D. Simmons-Duffin, Scaling the semidefinite program solver SDPB, arXiv:1909.09745 [INSPIRE].
M. Go and Y. Tachikawa, autoboot: a generator of bootstrap equations with global symmetry, JHEP 06 (2019) 084 [arXiv:1903.10522] [INSPIRE].
Z. Komargodski and D. Simmons-Duffin, The random-bond Ising model in 2.01 and 3 dimensions, J. Phys. A 50 (2017) 154001 [arXiv:1603.04444] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2006.01847
Rights and permissions
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.
About this article
Cite this article
Gimenez-Grau, A., Liendo, P. Bootstrapping Coulomb and Higgs branch operators. J. High Energ. Phys. 2021, 175 (2021). https://doi.org/10.1007/JHEP01(2021)175
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2021)175