Abstract
We study 2d N=4 superconformal field theories, focusing on its application on numerical bootstrap study. We derive the superconformal block by utilizing the global part of the super Virasoro algebra and set up the crossing equations for the non-BPS long-multiplet 4-point function. Along the way, we build global N=4 superconformal short and long multiplets and compute all possible 2,3-point functions of long-multiplets that are needed to construct the superconformal blocks and the crossing equations. Since we consider a long-multiplet 4-point function, the number of crossing equations is huge, and we expect it to give a strong constraint than the usual superconformal bootstrap analysis, which relies on BPS 4-point functions. In addition, we present an alternative way to derive crossing equations using N=4 superspace and comment on a puzzle.
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References
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].
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, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N) Archipelago, JHEP 11 (2015) 106 [arXiv:1504.07997] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [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].
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, E. Lauria and D. Mazac, Superconformal Blocks for SCFTs with Eight Supercharges, JHEP 07 (2017) 061 [arXiv:1705.08594] [INSPIRE].
M. Cornagliotto, M. Lemos and V. Schomerus, Long Multiplet Bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [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].
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, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The \( \mathcal{N}=2 \) superconformal bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [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].
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].
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].
M. Ademollo et al., Supersymmetric Strings and Color Confinement, Phys. Lett. 62B (1976) 105 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
V.A. Belavin, N = 1 supersymmetric conformal block recursion relations, Theor. Math. Phys. 152 (2007) 1275 [hep-th/0611295] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursion representation of the Neveu-Schwarz superconformal block, JHEP 03 (2007) 032 [hep-th/0611266] [INSPIRE].
O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
J. Kim, S. Kim, K. Lee, J. Park and C. Vafa, Elliptic Genus of E-strings, JHEP 09 (2017) 098 [arXiv:1411.2324] [INSPIRE].
E. Witten, On the conformal field theory of the Higgs branch, JHEP 07 (1997) 003 [hep-th/9707093] [INSPIRE].
A. Kapustin, Holomorphic reduction of N = 2 gauge theories, Wilson-’t Hooft operators and S-duality, hep-th/0612119 [INSPIRE].
P. Putrov, J. Song and W. Yan, (0, 4) dualities, JHEP 03 (2016) 185 [arXiv:1505.07110] [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, arXiv:0907.3987 [INSPIRE].
A. Hanany and T. Okazaki, (0, 4) brane box models, arXiv:1811.09117 [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, L. Rastelli and B.C. van Rees, More \( \mathcal{N}=4 \) superconformal bootstrap, Phys. Rev. D 96 (2017) 046014 [arXiv:1612.02363] [INSPIRE].
F. Bonetti, C. Meneghelli and L. Rastelli, VOAs labelled by complex reflection groups and 4d SCFTs, arXiv:1810.03612 [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].
P. Bouwknegt, Extended conformal algebras, Phys. Lett. B 207 (1988) 295 [INSPIRE].
K. Schoutens, O(n) Extended Superconformal Field Theory in Superspace, Nucl. Phys. B 295 (1988) 634 [INSPIRE].
T. Eguchi and A. Taormina, Unitary Representations of N = 4 Superconformal Algebra, Phys. Lett. B 196 (1987) 75 [INSPIRE].
T. Eguchi and A. Taormina, On the Unitary Representations of N = 2 and N = 4 Superconformal Algebras, Phys. Lett. B 210 (1988) 125 [INSPIRE].
T. Eguchi and A. Taormina, Character Formulas for the N = 4 Superconformal Algebra, Phys. Lett. B 200 (1988) 315 [INSPIRE].
M. Headrick, http://people.brandeis.edu/~headrick/Mathematica.
S. Matsuda and T. Uematsu, Chiral Superspace Formulation of N = 4 Superconformal Algebras, Phys. Lett. B 220 (1989) 413 [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Deformations of Superconformal Theories, JHEP 11 (2016) 135 [arXiv:1602.01217] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of Superconformal Symmetry in Diverse Dimensions, arXiv:1612.00809 [INSPIRE].
A.L. 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].
J. Murugan, D. Stanford and E. Witten, More on Supersymmetric and 2d Analogs of the SYK Model, JHEP 08 (2017) 146 [arXiv:1706.05362] [INSPIRE].
D. Simmons-Duffin, The Conformal Bootstrap, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), Boulder, CO, U.S.A., June 1-26, 2015, pp. 1-74 (2017) [https://doi.org/10.1142/9789813149441_0001] [arXiv:1602.07982] [INSPIRE].
D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
A. Dymarsky, F. Kos, P. Kravchuk, D. Poland and D. Simmons-Duffin, The 3d Stress-Tensor Bootstrap, JHEP 02 (2018) 164 [arXiv:1708.05718] [INSPIRE].
O. Ganor, unpublished notes.
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Kos, F., Oh, J. 2d small N=4 Long-multiplet superconformal block. J. High Energ. Phys. 2019, 1 (2019). https://doi.org/10.1007/JHEP02(2019)001
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DOI: https://doi.org/10.1007/JHEP02(2019)001