Abstract
We study large-c SCFT2 on a torus specializing to one-point superblocks in the \( \mathcal{N} \) = 1 Neveu-Schwarz sector. Considering different contractions of the Neveu-Schwarz superalgebra related to the large central charge limit we explicitly calculate three superblocks, osp(1|2) global, light, and heavy-light superblocks, and show that they are related to each other. We formulate the osp(1|2) superCasimir eigenvalue equations and identify their particular solutions as the global superblocks. It is shown that the resulting differential equations are the Heun equations. We study exponentiated global superblocks arising at large conformal dimensions and demonstrate that in the leading approximation the osp(1|2) superblocks are equal to the non-supersymmetric sl(2) block.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
P. Caputa, J. Simón, A. Štikonas and T. Takayanagi, Quantum Entanglement of Localized Excited States at Finite Temperature, JHEP 01 (2015) 102 [arXiv:1410.2287] [INSPIRE].
J. de Boer, A. Castro, E. Hijano, J.I. Jottar and P. Kraus, Higher spin entanglement and \( {\mathcal{W}}_{\mathrm{N}} \) conformal blocks, JHEP 07 (2015) 168 [arXiv:1412.7520] [INSPIRE].
E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP 07 (2015) 131 [arXiv:1501.02260] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, Classical conformal blocks via AdS/CFT correspondence, JHEP 08 (2015) 049 [arXiv:1504.05943] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS 3 gravity, JHEP 12 (2015) 077 [arXiv:1508.04987] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, Monodromic vs geodesic computation of Virasoro classical conformal blocks, Nucl. Phys. B 904 (2016) 367 [arXiv:1510.06685] [INSPIRE].
M. Beccaria, A. Fachechi and G. Macorini, Virasoro vacuum block at next-to-leading order in the heavy-light limit, JHEP 02 (2016) 072 [arXiv:1511.05452] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Conformal Blocks Beyond the Semi-Classical Limit, JHEP 05 (2016) 075 [arXiv:1512.03052] [INSPIRE].
P. Banerjee, S. Datta and R. Sinha, Higher-point conformal blocks and entanglement entropy in heavy states, JHEP 05 (2016) 127 [arXiv:1601.06794] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, Holographic interpretation of 1-point toroidal block in the semiclassical limit, JHEP 06 (2016) 183 [arXiv:1603.08440] [INSPIRE].
B. Chen, J.-q. Wu and J.-j. Zhang, Holographic Description of 2D Conformal Block in Semi-classical Limit, JHEP 10 (2016) 110 [arXiv:1609.00801] [INSPIRE].
K.B. Alkalaev, Many-point classical conformal blocks and geodesic networks on the hyperbolic plane, JHEP 12 (2016) 070 [arXiv:1610.06717] [INSPIRE].
P. Kraus and A. Maloney, A cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE].
O. Hulík, T. Procházka and J. Raeymaekers, Multi-centered AdS 3 solutions from Virasoro conformal blocks, JHEP 03 (2017) 129 [arXiv:1612.03879] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Exact Virasoro Blocks from Wilson Lines and Background-Independent Operators, JHEP 07 (2017) 092 [arXiv:1612.06385] [INSPIRE].
P. Kraus, A. Maloney, H. Maxfield, G.S. Ng and J.-q. Wu, Witten Diagrams for Torus Conformal Blocks, JHEP 09 (2017) 149 [arXiv:1706.00047] [INSPIRE].
V.A. Belavin and R.V. Geiko, Geodesic description of Heavy-Light Virasoro blocks, JHEP 08 (2017) 125 [arXiv:1705.10950] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, Holographic duals of large-c torus conformal blocks, JHEP 10 (2017) 140 [arXiv:1707.09311] [INSPIRE].
H. Maxfield, A view of the bulk from the worldline, arXiv:1712.00885 [INSPIRE].
Y. Kusuki, New Properties of Large-c Conformal Blocks from Recursion Relation, JHEP 07 (2018) 010 [arXiv:1804.06171] [INSPIRE].
E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, arXiv:1804.07924 [INSPIRE].
A. Belavin, V. Belavin, A. Neveu and A. Zamolodchikov, Bootstrap in Supersymmetric Liouville Field Theory. I. NS Sector, Nucl. Phys. B 784 (2007) 202 [hep-th/0703084] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, From global to heavy-light: 5-point conformal blocks, JHEP 03 (2016) 184 [arXiv:1512.07627] [INSPIRE].
K.B. Alkalaev, R.V. Geiko and V.A. Rappoport, Various semiclassical limits of torus conformal blocks, JHEP 04 (2017) 070 [arXiv:1612.05891] [INSPIRE].
M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, arXiv:1703.09805 [INSPIRE].
I.P. Ennes, A.V. Ramallo and J.M. Sanchez de Santos, OSP (1|2) conformal field theory, AIP Conf. Proc. 419 (1998) 138 [hep-th/9708094] [INSPIRE].
G. Götz, T. Quella and V. Schomerus, Representation theory of sl(2|1), J. Algebra 312 (2007) 829 [hep-th/0504234] [INSPIRE].
A. Lesniewski, A remark on the Casimir elements of Lie superalgebras and quantized Lie superalgebras, J. Math. Phys. 36 (1995) 1457.
D. Arnaudon and M. Bauer, Scasimir operator, scentre and representations of U q(osp(1|2)), Lett. Math. Phys. 40 (1997) 307 [q-alg/9605020] [INSPIRE].
P.K. Ghosh, SuperCalogero model with OSp(2|2) supersymmetry: Is the construction unique?, Nucl. Phys. B 681 (2004) 359 [hep-th/0309183] [INSPIRE].
D. Friedan, Z.-a. Qiu and S.H. Shenker, Superconformal Invariance in Two-Dimensions and the Tricritical Ising Model, Phys. Lett. B 151 (1985) 37 [INSPIRE].
M.A. Bershadsky, V.G. Knizhnik and M.G. Teitelman, Superconformal Symmetry in Two-Dimensions, Phys. Lett. B 151 (1985) 31 [INSPIRE].
L. Álvarez-Gaumé and P. Zaugg, Structure constants in the N = 1 superoperator algebra, Annals Phys. 215 (1992) 171 [hep-th/9109050] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recurrence relations for toric N = 1 superconformal blocks, JHEP 09 (2012) 122 [arXiv:1207.5740] [INSPIRE].
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE].
M. Piatek, Classical torus conformal block, N = 2∗ twisted superpotential and the accessory parameter of Lamé equation, JHEP 03 (2014) 124 [arXiv:1309.7672] [INSPIRE].
P. Menotti, Torus classical conformal blocks, arXiv:1805.07788 [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [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].
N. Bobev, S. El-Showk, D. Mazac and M.F. Paulos, Bootstrapping SCFTs with Four Supercharges, JHEP 08 (2015) 142 [arXiv:1503.02081] [INSPIRE].
M. Cornagliotto, M. Lemos and V. Schomerus, Long Multiplet Bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [INSPIRE].
D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
Y. Gobeil, A. Maloney, G.S. Ng and J.-q. Wu, Thermal Conformal Blocks, arXiv:1802.10537 [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques and Applications, arXiv:1805.04405 [INSPIRE].
T. Oshima, Fractional Calculus of Weyl Algebra and Fuchsian Differential Equations, Math. Soc. Japan Memoirs 28 (2012) 1.
A. Litvinov, S. Lukyanov, N. Nekrasov and A. Zamolodchikov, Classical Conformal Blocks and Painleve VI, JHEP 07 (2014) 144 [arXiv:1309.4700] [INSPIRE].
M. Piatek and A.R. Pietrykowski, Solving Heun’s equation using conformal blocks, arXiv:1708.06135 [INSPIRE].
M. Lencsés and F. Novaes, Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations, JHEP 04 (2018) 096 [arXiv:1709.03476] [INSPIRE].
H. Chen, A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Degenerate Operators and the 1/c Expansion: Lorentzian Resummations, High Order Computations and Super-Virasoro Blocks, JHEP 03 (2017) 167 [arXiv:1606.02659] [INSPIRE].
H. Poghosyan, The light asymptotic limit of conformal blocks in \( \mathcal{N} \) = 1 super Liouville field theory, JHEP 09 (2017) 062 [arXiv:1706.07474] [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].
P. Goddard, A. Kent and D.I. Olive, Unitary Representations of the Virasoro and Supervirasoro Algebras, Commun. Math. Phys. 103 (1986) 105 [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Elliptic recurrence representation of the N = 1 Neveu-Schwarz blocks, Nucl. Phys. B 798 (2008) 363 [arXiv:0711.1619] [INSPIRE].
M. Nishida and K. Tamaoka, Fermions in Geodesic Witten Diagrams, JHEP 07 (2018) 149 [arXiv:1805.00217] [INSPIRE].
D. Friedan, Notes on string theory and two-dimensional conformal field theory, preprint EFI 85-99 (1986) [INSPIRE].
M. Dorrzapf, The Definition of Neveu-Schwarz superconformal fields and uncharged superconformal transformations, Rev. Math. Phys. 11 (1999) 137 [hep-th/9712107] [INSPIRE].
H. Bateman and A. Erdelyi, Higher Transcendental Functions, Volume 3, McGraw-Hill (1953).
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: 1805.12585
Weston Visiting Professorship at Weizmann Institute (Vladimir Belavin).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Alkalaev, K., Belavin, V. Large-c superconformal torus blocks. J. High Energ. Phys. 2018, 42 (2018). https://doi.org/10.1007/JHEP08(2018)042
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)042