Constraints on higher spin \(\mathrm{CFT}_{2}\). (English) Zbl 1391.83083
Summary: We derive constraints on two-dimensional conformal field theories with higher spin symmetry due to unitarity, modular invariance, and causality. We focus on CFTs with \( {\mathcal{W}}_N \) symmetry in the “irrational” regime, where \(c>N-1\) and the theories have an infinite number of higher-spin primaries. The most powerful constraints come from positivity of the Kac matrix, which (unlike the Virasoro case) is non-trivial even when \( c > N - 1\). This places a lower bound on the dimension of any non-vacuum higher-spin primary state, which is linear in the central charge. At large \(c\), this implies that the dual holographic theories of gravity in \(\mathrm{AdS}_{3}\), if they exist, have no local, perturbative degrees of freedom in the semi-classical limit.
MSC:
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
Keywords:
higher spin symmetry; conformal field theory; field theories in lower dimensions; higher spin gravitySoftware:
MathematicaReferences:
| [1] | Belavin, AA; Polyakov, AM; Zamolodchikov, AB, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys., B 241, 333, (1984) · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X |
| [2] | Cappelli, A.; Itzykson, C.; Zuber, JB, Modular invariant partition functions in two-dimensions, Nucl. Phys., B 280, 445, (1987) · Zbl 0661.17017 · doi:10.1016/0550-3213(87)90155-6 |
| [3] | Asplund, CT; Bernamonti, A.; Galli, F.; Hartman, T., entanglement scrambling in 2d conformal field theory, JHEP, 09, 110, (2015) · Zbl 1388.83165 · doi:10.1007/JHEP09(2015)110 |
| [4] | A. Bilal, Introduction to W algebras, in proceedings of Spring School on String Theory and Quantum Gravity, Trieste, Italy, April 15-23, 1991, pp. 245-280 [INSPIRE]. |
| [5] | Bouwknegt, P.; Schoutens, K., W symmetry in conformal field theory, Phys. Rept., 223, 183, (1993) · doi:10.1016/0370-1573(93)90111-P |
| [6] | Maldacena, J.; Zhiboedov, A., Constraining conformal field theories with A higher spin symmetry, J. Phys., A 46, 214011, (2013) · Zbl 1339.81089 |
| [7] | Boulanger, N.; Ponomarev, D.; Skvortsov, ED; Taronna, M., On the uniqueness of higher-spin symmetries in AdS and CFT, Int. J. Mod. Phys., A 28, 1350162, (2013) · Zbl 1284.81250 · doi:10.1142/S0217751X13501625 |
| [8] | Alba, V.; Diab, K., constraining conformal field theories with a higher spin symmetry in d > 3 dimensions, JHEP, 03, 044, (2016) · doi:10.1007/JHEP03(2016)044 |
| [9] | Hellerman, S., A universal inequality for CFT and quantum gravity, JHEP, 08, 130, (2011) · Zbl 1298.83051 · doi:10.1007/JHEP08(2011)130 |
| [10] | Friedan, D.; Keller, CA, constraints on 2d CFT partition functions, JHEP, 10, 180, (2013) · Zbl 1342.81361 · doi:10.1007/JHEP10(2013)180 |
| [11] | Qualls, JD; Shapere, AD, Bounds on operator dimensions in 2D conformal field theories, JHEP, 05, 091, (2014) · doi:10.1007/JHEP05(2014)091 |
| [12] | Benjamin, N.; Dyer, E.; Fitzpatrick, AL; Kachru, S., universal bounds on charged states in 2d CFT and 3d gravity, JHEP, 08, 041, (2016) · Zbl 1390.83244 · doi:10.1007/JHEP08(2016)041 |
| [13] | S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, arXiv:1608.06241 [INSPIRE]. · Zbl 1044.81594 |
| [14] | L. Apolo, Bounds on CFTs with\( {\mathcal{W}}_3 \)algebras and AdS_{3}higher spin theories, Phys. Rev.D 96 (2017) 086003 [arXiv:1705.10402] [INSPIRE]. · Zbl 1002.17014 |
| [15] | Iles, NJ; Watts, GMT, characters of the W_{3}algebra, JHEP, 02, 009, (2014) · Zbl 1333.83191 · doi:10.1007/JHEP02(2014)009 |
| [16] | Iles, NJ; Watts, GMT, modular properties of characters of the W_{3}algebra, JHEP, 01, 089, (2016) · Zbl 1388.81220 · doi:10.1007/JHEP01(2016)089 |
| [17] | Perlmutter, E., Bounding the space of holographic CFTs with chaos, JHEP, 10, 069, (2016) · Zbl 1390.83128 · doi:10.1007/JHEP10(2016)069 |
| [18] | E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE]. · Zbl 0768.53042 |
| [19] | E. Witten, (2 + 1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys.B 311 (1988) 46 [INSPIRE]. · Zbl 1258.83032 |
| [20] | Achucarro, A.; Townsend, PK, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett., B 180, 89, (1986) · doi:10.1016/0370-2693(86)90140-1 |
| [21] | Campoleoni, A.; Fredenhagen, S.; Pfenninger, S.; Theisen, S., Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP, 11, 007, (2010) · Zbl 1294.81240 · doi:10.1007/JHEP11(2010)007 |
| [22] | Gaberdiel, MR; Gopakumar, R., an AdS_{3}dual for minimal model CFTs, Phys. Rev., D 83, (2011) |
| [23] | Chang, C-M; Yin, X., higher spin gravity with matter in AdS_{3}and its CFT dual, JHEP, 10, 024, (2012) · Zbl 1397.83132 · doi:10.1007/JHEP10(2012)024 |
| [24] | Papadodimas, K.; Raju, S., Correlation functions in holographic minimal models, Nucl. Phys., B 856, 607, (2012) · Zbl 1246.81177 · doi:10.1016/j.nuclphysb.2011.11.006 |
| [25] | Gaberdiel, MR; Gopakumar, R.; Rangamani, M., The spectrum of light states in large N minimal models, JHEP, 01, 116, (2014) · Zbl 1333.81445 · doi:10.1007/JHEP01(2014)116 |
| [26] | Castro, A.; Lepage-Jutier, A.; Maloney, A., higher spin theories in AdS_{3}and a gravitational exclusion principle, JHEP, 01, 142, (2011) · Zbl 1214.83008 · doi:10.1007/JHEP01(2011)142 |
| [27] | Cardy, JL, Operator content of two-dimensional conformally invariant theories, Nucl. Phys., B 270, 186, (1986) · Zbl 0689.17016 · doi:10.1016/0550-3213(86)90552-3 |
| [28] | Komargodski, Z.; Zhiboedov, A., Convexity and liberation at large spin, JHEP, 11, 140, (2013) · doi:10.1007/JHEP11(2013)140 |
| [29] | Fitzpatrick, AL; Kaplan, J.; Poland, D.; Simmons-Duffin, D., The analytic bootstrap and AdS superhorizon locality, JHEP, 12, 004, (2013) · Zbl 1342.83239 · doi:10.1007/JHEP12(2013)004 |
| [30] | Hartman, T.; Keller, CA; Stoica, B., Universal spectrum of 2d conformal field theory in the large c limit, JHEP, 09, 118, (2014) · Zbl 1333.81368 · doi:10.1007/JHEP09(2014)118 |
| [31] | Maldacena, J.; Shenker, SH; Stanford, D., A bound on chaos, JHEP, 08, 106, (2016) · Zbl 1390.81388 · doi:10.1007/JHEP08(2016)106 |
| [32] | Hartman, T.; Jain, S.; Kundu, S., Causality constraints in conformal field theory, JHEP, 05, 099, (2016) · doi:10.1007/JHEP05(2016)099 |
| [33] | Ferrara, S.; Gatto, R.; Grillo, AF, Properties of partial wave amplitudes in conformal invariant field theories, Nuovo Cim., A 26, 226, (1975) · doi:10.1007/BF02769009 |
| [34] | Fateev, V.; Ribault, S., The large central charge limit of conformal blocks, JHEP, 02, 001, (2012) · Zbl 1309.81154 · doi:10.1007/JHEP02(2012)001 |
| [35] | Gaberdiel, MR; Hartman, T., Symmetries of holographic minimal models, JHEP, 05, 031, (2011) · Zbl 1296.81093 · doi:10.1007/JHEP05(2011)031 |
| [36] | Gaberdiel, MR; Hartman, T.; Jin, K., Higher spin black holes from CFT, JHEP, 04, 103, (2012) · Zbl 1348.83079 · doi:10.1007/JHEP04(2012)103 |
| [37] | Thielemans, K., A Mathematica package for computing operator product expansions, Int. J. Mod. Phys., C 2, 787, (1991) · Zbl 0940.81500 · doi:10.1142/S0129183191001001 |
| [38] | Mizoguchi, S., Nonunitarity theorem for the a type W_{n} algebra, Phys. Lett., B 231, 112, (1989) · doi:10.1016/0370-2693(89)90122-6 |
| [39] | G.M.T. Watts, Determinant Formulae for Extended Algebras in Two-dimensional Conformal Field Theory, Nucl. Phys.B 326 (1989) 648 [Erratum ibid.B 336 (1990) 720] [INSPIRE]. · Zbl 1333.81445 |
| [40] | Zamolodchikov, AB, Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Math. Phys., 65, 1205, (1985) · doi:10.1007/BF01036128 |
| [41] | R. Blumenhagen and E. Plauschinn, Introduction to Conformal Field Theory: With Applications to String Theory, Lect. Notes Phys., Springer Berlin Heidelberg (2009). · Zbl 1175.81001 |
| [42] | Mizoguchi, S., determinant formula and unitarity for the W_{3}algebra, Phys. Lett., B 222, 226, (1989) · doi:10.1016/0370-2693(89)91256-2 |
| [43] | Blumenhagen, R.; Flohr, M.; Kliem, A.; Nahm, W.; Recknagel, A.; Varnhagen, R., W algebras with two and three generators, Nucl. Phys., B 361, 255, (1991) · doi:10.1016/0550-3213(91)90624-7 |
| [44] | Kausch, HG; Watts, GMT, A study of W algebras using Jacobi identities, Nucl. Phys., B 354, 740, (1991) · doi:10.1016/0550-3213(91)90375-8 |
| [45] | Zhu, C-J, the complete structure of the nonlinear W_{4}and W_{5}algebras from quantum miura transformation, Phys. Lett., B 316, 264, (1993) · doi:10.1016/0370-2693(93)90324-B |
| [46] | Hornfeck, K., W algebras with set of primary fields of dimensions (3\(,\) 4\(,\) 5) and (3\(,\) 4\(,\) 5\(,\) 6), Nucl. Phys., B 407, 237, (1993) · Zbl 1043.81566 · doi:10.1016/0550-3213(93)90281-S |
| [47] | J.D. Qualls, Universal Bounds on Operator Dimensions in General 2D Conformal Field Theories, arXiv:1508.00548 [INSPIRE]. · Zbl 1390.83128 |
| [48] | Vos, K.; Driel, P., The Kazhdan-Lusztig conjecture for W algebras, J. Math. Phys., 37, 3587, (1996) · Zbl 0862.17020 · doi:10.1063/1.531584 |
| [49] | Mizoguchi, S., the structure of representation of the W_{3}algebra, Int. J. Mod. Phys., A 6, 133, (1991) · Zbl 0736.17036 · doi:10.1142/S0217751X91000125 |
| [50] | Borisov, L.; Halpern, MB; Schweigert, C., Systematic approach to cyclic orbifolds, Int. J. Mod. Phys., A 13, 125, (1998) · Zbl 1002.17014 · doi:10.1142/S0217751X98000044 |
| [51] | Blumenhagen, R.; Eholzer, W.; Honecker, A.; Hornfeck, K.; Hubel, R., Coset realization of unifying W algebras, Int. J. Mod. Phys., A 10, 2367, (1995) · Zbl 1044.81594 · doi:10.1142/S0217751X95001157 |
| [52] | P. Bouwknegt, Extended Conformal Algebras from Kac-Moody Algebras, (1988) [INSPIRE]. · Zbl 0744.17031 |
| [53] | Odake, S., \(C\) = 3d conformal algebra with extended supersymmetry, Mod. Phys. Lett., A 5, 561, (1990) · Zbl 1020.81633 · doi:10.1142/S0217732390000640 |
| [54] | Odake, S., Character formulas of an extended superconformal algebra relevant to string compactification, Int. J. Mod. Phys., A 5, 897, (1990) · doi:10.1142/S0217751X90000428 |
| [55] | Odake, S., extension of N = 2 superconformal algebra and Calabi-Yau compactification, Mod. Phys. Lett., A 4, 557, (1989) · doi:10.1142/S021773238900068X |
| [56] | Shatashvili, SL; Vafa, C., Superstrings and manifold of exceptional holonomy, Selecta Math., 1, 347, (1995) · Zbl 0839.53060 · doi:10.1007/BF01671569 |
| [57] | Gepner, D.; Noyvert, B., unitary representations of SW (3\(/\)2\(,\) 2) superconformal algebra, Nucl. Phys., B 610, 545, (2001) · Zbl 0971.17014 · doi:10.1016/S0550-3213(01)00299-1 |
| [58] | Benjamin, N.; Harrison, SM; Kachru, S.; Paquette, NM; Whalen, D., on the elliptic genera of manifolds of spin(7) holonomy, Annales Henri Poincaré, 17, 2663, (2016) · Zbl 1350.81026 · doi:10.1007/s00023-015-0454-5 |
| [59] | Mallwitz, S., on SW minimal models and N = 1 supersymmetric quantum Toda field theories, Int. J. Mod. Phys., A 10, 977, (1995) · Zbl 1044.81601 · doi:10.1142/S0217751X95000486 |
| [60] | Noyvert, B., unitary minimal models of SW (3\(/\)2\(,\) 3\(/\)2\(,\) 2) superconformal algebra and manifolds of G_{2}holonomy, JHEP, 03, 030, (2002) · doi:10.1088/1126-6708/2002/03/030 |
| [61] | Boer, J.; Naqvi, A.; Shomer, A., the topological G_{2}string, Adv. Theor. Math. Phys., 12, 243, (2008) · Zbl 1144.81502 · doi:10.4310/ATMP.2008.v12.n2.a2 |
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