Abstract
We use the conformal bootstrap to perform a precision study of 3d maximally supersymmetric (\( \mathcal{N}=8 \)) SCFTs that describe the IR physics on N coincident M2-branes placed either in flat space or at a ℂ4/ℤ2 singularity. First, using the explicit Lagrangians of ABJ(M) [1, 2] and recent supersymmetric localization results, we calculate certain half and quarter-BPS OPE coefficients, both exactly at small N, and approximately in a large N expansion that we perform to all orders in 1/N. Comparing these values with the numerical bootstrap bounds leads us to conjecture that some of these theories obey an OPE coefficient minimization principle. We then use this conjecture as well as the extremal functional method to reconstruct the first few low-lying scaling dimensions and OPE coefficients for both protected and unprotected multiplets that appear in the OPE of two stress tensor multiplets for all values of N. We also calculate the half and quarter-BPS operator OPE coefficients in the SU(2)k × SU(2)−k BLG theory for all values of the Chern-Simons coupling k, and show that generically they do not obey the same OPE coefficient minimization principle.
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O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].
A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
G. Mack, Duality in quantum field theory, Nucl. Phys. B 118 (1977) 445 [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].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, JHEP 11 (2014) 109 [arXiv:1406.4858] [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].
V.S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
A. Vichi, Improved bounds for CFT’s with global symmetries, JHEP 01 (2012) 162 [arXiv:1106.4037] [INSPIRE].
D. Poland, D. Simmons-Duffin and A. Vichi, Carving out the space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFT d, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
S. El-Showk et al., Conformal field theories in fractional dimensions, Phys. Rev. Lett. 112 (2014) 141601 [arXiv:1309.5089] [INSPIRE].
D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
S.M. Chester, S.S. Pufu and R. Yacoby, Bootstrapping O(N) vector models in 4 < d < 6, Phys. Rev. D 91 (2015) 086014 [arXiv:1412.7746] [INSPIRE].
C. Beem et al., The \( \mathcal{N}=2 \) superconformal bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE].
S.M. Chester et al., Accidental symmetries and the conformal bootstrap, JHEP 01 (2016) 110 [arXiv:1507.04424] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
D. Poland and A. Stergiou, Exploring the minimal 4D \( \mathcal{N}=1 \) SCFT, JHEP 12 (2015) 121 [arXiv:1509.06368] [INSPIRE].
M. Lemos and P. Liendo, Bootstrapping \( \mathcal{N}=2 \) chiral correlators, JHEP 01 (2016) 025 [arXiv:1510.03866] [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].
Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, \( \mathcal{N}=4 \) superconformal bootstrap of the K3 CFT, JHEP 05 (2017) 126 [arXiv:1511.04065] [INSPIRE].
S.M. Chester and S.S. Pufu, Towards bootstrapping QED 3, JHEP 08 (2016) 019 [arXiv:1601.03476] [INSPIRE].
M. Lemos, P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping \( \mathcal{N}=3 \) superconformal theories, JHEP 04 (2017) 032 [arXiv:1612.01536] [INSPIRE].
L. Iliesiu et al., Bootstrapping 3D fermions with global symmetries, JHEP 01 (2018) 036 [arXiv:1705.03484] [INSPIRE].
M. Cornagliotto, M. Lemos and V. Schomerus, Long multiplet bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [INSPIRE].
D. Li, D. Meltzer and A. Stergiou, Bootstrapping mixed correlators in 4D \( \mathcal{N}=1 \) SCFTs, JHEP 07 (2017) 029 [arXiv:1702.00404] [INSPIRE].
A. Dymarsky, J. Penedones, E. Trevisani and A. Vichi, Charting the space of 3D CFTs with a continuous global symmetry, arXiv:1705.04278 [INSPIRE].
S.M. Chester, L.V. Iliesiu, M. Mezei and S.S. Pufu, Monopole operators in U(1) Chern-Simons-Matter theories, JHEP 05 (2018) 157 [arXiv:1710.00654] [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].
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].
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].
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].
S. El-Showk et al., 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, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
S. El-Showk and M.F. Paulos, Extremal bootstrapping: go with the flow, JHEP 03 (2018) 148 [arXiv:1605.08087] [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].
N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping the three-dimensional supersymmetric Ising model, Phys. Rev. Lett. 115 (2015) 051601 [arXiv:1502.04124] [INSPIRE].
M. Van Raamsdonk, Comments on the Bagger-Lambert theory and multiple M2-branes, JHEP 05 (2008) 105 [arXiv:0803.3803] [INSPIRE].
M.A. Bandres, A.E. Lipstein and J.H. Schwarz, N = 8 superconformal Chern-Simons theories, JHEP 05 (2008) 025 [arXiv:0803.3242] [INSPIRE].
J. Bagger and N. Lambert, Comments on multiple M2-branes, JHEP 02 (2008) 105 [arXiv:0712.3738] [INSPIRE].
J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [INSPIRE].
J. Bagger and N. Lambert, Modeling multiple M2’s, Phys. Rev. D 75 (2007) 045020 [hep-th/0611108] [INSPIRE].
A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66 [arXiv:0709.1260] [INSPIRE].
D. Bashkirov and A. Kapustin, Dualities between N = 8 superconformal field theories in three dimensions, JHEP 05 (2011) 074 [arXiv:1103.3548] [INSPIRE].
N.B. Agmon, S.M. Chester and S.S. Pufu, A new duality between \( \mathcal{N}=8 \) superconformal field theories in three dimensions, JHEP 06 (2018) 005 [arXiv:1708.07861] [INSPIRE].
N. Lambert and C. Papageorgakis, Relating U(N) × U(N) to SU(N) × SU(N) Chern-Simons Membrane theories, JHEP 04 (2010) 104 [arXiv:1001.4779] [INSPIRE].
D. Bashkirov, BLG theories at low values of Chern-Simons coupling, arXiv:1211.4887 [INSPIRE].
S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, The \( \mathcal{N}=8 \) 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].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric field theories on three-manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].
Y. Imamura and D. Yokoyama, N = 2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].
C. Beem, W. Peelaers and L. Rastelli, Deformation quantization and superconformal symmetry in three dimensions, Commun. Math. Phys. 354 (2017) 345 [arXiv:1601.05378] [INSPIRE].
C. Beem et al., Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
M. Dedushenko, S.S. Pufu and R. Yacoby, A one-dimensional theory for Higgs branch operators, JHEP 03 (2018) 138 [arXiv:1610.00740] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
T. Nosaka, Instanton effects in ABJM theory with general R-charge assignments, JHEP 03 (2016) 059 [arXiv:1512.02862] [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 783 [hep-th/9712074] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju, Indices for superconformal field theories in 3, 5 and 6 dimensions, JHEP 02 (2008) 064 [arXiv:0801.1435] [INSPIRE].
F.A. Dolan, On superconformal characters and partition functions in three dimensions, J. Math. Phys. 51 (2010) 022301 [arXiv:0811.2740] [INSPIRE].
S. Ferrara and E. Sokatchev, Universal properties of superconformal OPEs for 1/2 BPS operators in 3 ≤ D ≤ 6, New J. Phys. 4 (2002) 2 [hep-th/0110174] [INSPIRE].
E. Gerchkovitz et al., Correlation Functions of Coulomb Branch Operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].
J. Gomis and N. Ishtiaque, Kähler potential and ambiguities in 4d \( \mathcal{N}=2 \) SCFTs, JHEP 04 (2015) 169 [arXiv:1409.5325] [INSPIRE].
C. Closset et al., Contact terms, unitarity and F-maximization in three-dimensional superconformal theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
C.P. Herzog, I.R. Klebanov, S.S. Pufu and T. Tesileanu, Multi-matrix models and Tri-Sasaki Einstein spaces, Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [INSPIRE].
M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [INSPIRE].
M. Honda and K. Okuyama, Exact results on ABJ theory and the refined topological string, JHEP 08 (2014) 148 [arXiv:1405.3653] [INSPIRE].
D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
O. Aharony, L.F. Alday, A. Bissi and E. Perlmutter, Loops in AdS from Conformal Field Theory, JHEP 07 (2017) 036 [arXiv:1612.03891] [INSPIRE].
D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem: N = 2 field theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE convergence in conformal field theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].
S. Rychkov and P. Yvernay, Remarks on the Convergence Properties of the Conformal Block Expansion, Phys. Lett. B 753 (2016) 682 [arXiv:1510.08486] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, The \( \mathcal{N}=4 \) superconformal bootstrap, Phys. Rev. Lett. 111 (2013) 071601 [arXiv:1304.1803] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, More \( \mathcal{N}=4 \) superconformal bootstrap, Phys. Rev. D 96 (2017) 046014 [arXiv:1612.02363] [INSPIRE].
M. Mariño and P. Putrov, Exact results in ABJM theory from topological strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].
Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton effects in ABJM theory from Fermi gas approach, JHEP 01 (2013) 158 [arXiv:1211.1251] [INSPIRE].
Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton bound states in ABJM theory, JHEP 05 (2013) 054 [arXiv:1301.5184] [INSPIRE].
F. Calvo and M. Mariño, Membrane instantons from a semiclassical TBA, JHEP 05 (2013) 006 [arXiv:1212.5118] [INSPIRE].
Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, JHEP 09 (2014) 168 [arXiv:1306.1734] [INSPIRE].
J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, Annales Henri Poincaré 17 (2016) 1037 [arXiv:1308.6485] [INSPIRE].
M. Honda, Direct derivation of “mirror” ABJ partition function, JHEP 12 (2013) 046 [arXiv:1310.3126] [INSPIRE].
S. Matsumoto and S. Moriyama, ABJ fractional brane from ABJM Wilson loop, JHEP 03 (2014) 079 [arXiv:1310.8051] [INSPIRE].
J. Kallen, The spectral problem of the ABJ Fermi gas, JHEP 10 (2015) 029 [arXiv:1407.0625] [INSPIRE].
S. Codesido, A. Grassi and M. Mariño, Exact results in \( \mathcal{N}=8 \) Chern-Simons-matter theories and quantum geometry, JHEP 07 (2015) 011 [arXiv:1409.1799] [INSPIRE].
S. Giombi et al., Chern-Simons theory with vector fermion matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].
C.-M. Chang, S. Minwalla, T. Sharma and X. Yin, ABJ Triality: from Higher Spin Fields to Strings, J. Phys. A 46 (2013) 214009 [arXiv:1207.4485] [INSPIRE].
M. Mezei, S.S. Pufu and Y. Wang, A 2d/1d holographic duality, arXiv:1703.08749 [INSPIRE].
A. Klemm et al., Aharony-Bergman-Jafferis-Maldacena Wilson loops in the Fermi gas approach, Z. Naturforsch. A 68 (2013) 178 [arXiv:1207.0611] [INSPIRE].
K. Okuyama, A note on the partition function of ABJM theory on S 3, Prog. Theor. Phys. 127 (2012) 229 [arXiv:1110.3555] [INSPIRE].
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Agmon, N.B., Chester, S.M. & Pufu, S.S. Solving M-theory with the conformal bootstrap. J. High Energ. Phys. 2018, 159 (2018). https://doi.org/10.1007/JHEP06(2018)159
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DOI: https://doi.org/10.1007/JHEP06(2018)159