Elliptic genera and 3d gravity. (English) Zbl 1351.83019
Summary: We describe general constraints on the elliptic genus of a 2d supersymmetric conformal field theory which has a gravity dual with large radius in Planck units. We give examples of theories which do and do not satisfy the bounds we derive, by describing the elliptic genera of symmetric product orbifolds of \(K3\), product manifolds, certain simple families of Calabi-Yau hypersurfaces, and symmetric products of the “Monster CFT”. We discuss the distinction between theories with supergravity duals and those whose duals have strings at the scale set by the AdS curvature. Under natural assumptions, we attempt to quantify the fraction of \((2,2)\) supersymmetric conformal theories which admit a weakly curved gravity description, at large central charge.
MSC:
| 83C45 | Quantization of the gravitational field |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 81V17 | Gravitational interaction in quantum theory |
| 83C80 | Analogues of general relativity in lower dimensions |
| 58D30 | Applications of manifolds of mappings to the sciences |
| 53Z05 | Applications of differential geometry to physics |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 32Q25 | Calabi-Yau theory (complex-analytic aspects) |
| 83E30 | String and superstring theories in gravitational theory |
| 83E50 | Supergravity |
Keywords:
elliptic genera; 3d gravity; constraints; 2d supersymmetric conformal field theory; orbifolds; Calabi-Yau hypersurfaces; supergravity; strings; AdS curvature; central chargeReferences:
| [1] | Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998). arXiv:hep-th/9711200 · Zbl 0914.53047 |
| [2] | Hartman, T., Keller, C. A., Stoica, B.: Universal spectrum of 2d conformal field theory in the large c limit. arXiv:1405.5137 [hep-th] · Zbl 1333.81368 |
| [3] | Witten, E.: Anti-de Sitter space, thermal phase transition, and confinement in gauge theories. Adv. Theor. Math. Phys. 2, 505 (1998). arXiv:hep-th/9803131 · Zbl 1057.81550 |
| [4] | Hawking S.W., Page D.N.: Thermodynamics of black holes in anti-De Sitter space. Commun. Math. Phys. 87, 577 (1983) · doi:10.1007/BF01208266 |
| [5] | Schellekens A.N., Warner N.P.: Anomalies and modular invariance in string theory. Phys. Lett. B 177, 317 (1986) · doi:10.1016/0370-2693(86)90760-4 |
| [6] | Schellekens A.N., Warner N.P.: Anomaly cancellation and selfdual lattices. Phys. Lett. B 181, 339 (1986) · doi:10.1016/0370-2693(86)90059-6 |
| [7] | Pilch K., Schellekens A.N., Warner N.P.: Path integral calculation of string anomalies. Nucl. Phys. B 287, 362 (1987) · Zbl 0635.53078 · doi:10.1016/0550-3213(87)90109-X |
| [8] | Witten E.: Elliptic genera and quantum field theory. Commun. Math. Phys. 109, 525 (1987) · Zbl 0625.57008 · doi:10.1007/BF01208956 |
| [9] | Eichler M., Zagier D.: The theory of Jacobi forms. Birkhaüser, Boston (1985) · Zbl 0554.10018 · doi:10.1007/978-1-4684-9162-3 |
| [10] | Strominger, A., Vafa, C.: Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. B 379, 99 (1996). arXiv:hep-th/9601029 · Zbl 1376.83026 |
| [11] | Kawai, T., Yamada, Y., Yang, S. K.: Elliptic genera and N =2 superconformal field theory. Nucl. Phys. B 414, 191 (1994). arXiv:hep-th/9306096 · Zbl 0980.58500 |
| [12] | Gaberdiel, M.R., Gukov, S., Keller, C.A., Moore, G.W., Ooguri, H.: Extremal N =(2,2) 2D conformal field theories and constraints of modularity. Commun. Number Theor. Phys. 2, 743 (2008), arXiv:0805.4216 [hep-th] · Zbl 1214.81245 |
| [13] | Dijkgraaf, R., Maldacena, J.M., Moore, G.W., Verlinde, E.P.: A black hole farey tail. arXiv:hep-th/0005003 · Zbl 0625.57008 |
| [14] | Manschot, J., Moore, G.W.: A modern farey tail. Commun. Number Theor. Phys. 4, 103 (2010), arXiv:0712.0573 [hep-th] · Zbl 1259.58005 |
| [15] | Aharony, O., Gubser, S. S., Maldacena, J.M., Ooguri, H., Oz, Y.: Large N field theories, string theory and gravity. Phys. Rep. 323, 183 (2000). arXiv:hep-th/9905111 · Zbl 1368.81009 |
| [16] | Kraus, P., Larsen, F.: Partition functions and elliptic genera from supergravity. JHEP 0701, 002 (2007). arXiv:hep-th/0607138 |
| [17] | Kraus, P.: Lectures on black holes and the AdS(3)/CFT(2) correspondence. Lect. Notes Phys. 755, 193 (2008). arXiv:hep-th/0609074 · Zbl 1155.83303 |
| [18] | Banados, M., Teitelboim, C., Zanelli, J.: The black hole in three-dimensional space-time. Phys. Rev. Lett. 69, 1849 (1992). arXiv:hep-th/9204099 · Zbl 0968.83514 |
| [19] | Cvetic, M., Larsen, F.: Near horizon geometry of rotating black holes in five-dimensions. Nucl. Phys. B 531, 239 (1998). arXiv:hep-th/9805097 · Zbl 0956.83033 |
| [20] | de Boer, J., Denef, F., El-Showk, S., Messamah, I., Van den Bleeken, D.: Black hole bound states in AdS(3) x S**2. JHEP 0811, 050 (2008). arXiv:0802.2257 [hep-th] |
| [21] | Bena, I., Chowdhury, B.D., de Boer, J., El-Showk, S., Shigemori, M.: moulting black holes. JHEP 1203, 094 (2012). arXiv:1108.0411 [hep-th] · Zbl 1309.83088 |
| [22] | Maldacena, J.M., Strominger, A.: AdS(3) black holes and a stringy exclusion principle. JHEP 9812, 005 (1998). arXiv:hep-th/9804085 · Zbl 0951.83019 |
| [23] | Maloney, A., Witten, E.: Quantum gravity partition functions in three dimensions. JHEP 1002, 029 (2010). arXiv:0712.0155 [hep-th] · Zbl 1270.83022 |
| [24] | Keller, C.A.: Phase transitions in symmetric orbifold CFTs and universality. JHEP 1103, 114 (2011). arXiv:1101.4937 [hep-th] · Zbl 1301.81223 |
| [25] | Dijkgraaf, R.: Instanton strings and hyperKahler geometry. Nucl. Phys. B 543, 545 (1999). arXiv:hep-th/9810210 · Zbl 1097.58503 |
| [26] | Dijkgraaf, R., Moore, G.W., Verlinde, E.P., Verlinde, H.L.: Elliptic genera of symmetric products and second quantized strings. Commun. Math. Phys. 185, 197 (1997). arXiv:hep-th/9608096 · Zbl 0872.32006 |
| [27] | Manschot, J., Zapata Rolon, J.M.: The asymptotic profile of χy-genera of Hilbert schemes of points on K3 surfaces. Commun. Number Theor. Phys. 09, 413 (2015). arXiv:1411.1093 [math.AG] · Zbl 1374.14034 |
| [28] | Maldacena, J.M., G.W., Strominger, A.: Counting BPS black holes in toroidal type II string theory. arXiv:hep-th/9903163 · Zbl 0951.83019 |
| [29] | Haehl, F.M., Rangamani, M.: Permutation orbifolds and holography. arXiv:1412.2759 [hep-th] · Zbl 1380.81369 |
| [30] | Belin, A., Keller, C.A., Maloney, A.: String universality for permutation orbifolds. arXiv:1412.7159 [hep-th] · Zbl 1357.81150 |
| [31] | Gritsenko, V.: Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms. arXiv:math/9906190 · Zbl 1101.14308 |
| [32] | Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the monster, Pure and Applied Mathematics, vol. 134. Academic (1988), Boston, p. 508 · Zbl 0674.17001 |
| [33] | Li, W., Song, W., Strominger, A.: Chiral gravity in three dimensions. JHEP 0804, 082 (2008). arXiv:0801.4566 [hep-th] · Zbl 1246.83158 |
| [34] | Maloney, A., Song, W., Strominger, A.: Chiral gravity, log gravity and extremal CFT. Phys. Rev. D 81, 064007 (2010). arXiv:0903.4573 [hep-th] |
| [35] | Skenderis, K., Taylor, M., Van Rees, B.C.: Topologically massive gravity and the AdS/CFT correspondence. JHEP 0909, 045 (2009). arXiv:0906.4926 [hep-th] |
| [36] | Borcherds R.E.: The monster Lie algebra. Adv. Math. 83, 30 (1990) · Zbl 0734.17010 · doi:10.1016/0001-8708(90)90067-W |
| [37] | de Boer, J.: Large N elliptic genus and AdS/CFT correspondence. JHEP 9905, 017 (1999). arXiv:hep-th/9812240 · Zbl 1017.81041 |
| [38] | de Boer, J., El-Showk, S., Messamah, I., Van den Bleeken, D.: A bound on the entropy of supergravity? JHEP 1002, 062 (2010). arXiv:0906.0011 [hep-th] · Zbl 1270.83024 |
| [39] | Minwalla, S., Narayan, P., Sharma, T., Umesh, V., Yin, X.: Supersymmetric states in large N Chern-Simons-Matter theories. JHEP 1202, 022 (2012). arXiv:1104.0680 [hep-th] · Zbl 1309.81166 |
| [40] | Zamolodchikov A.B.: Irreversibility of the flux of the renormalization group in a 2D field theory. JETP Lett. 43, 730 (1986) |
| [41] | Horne, J.H., Moore, G.W.: Chaotic coupling constants. Nucl. Phys. B 432, 109 (1994). arXiv:hep-th/9403058 · Zbl 1020.81780 |
| [42] | Douglas, M.R., Lu, Z.: Finiteness of volume of moduli spaces. arXiv:hep-th/0509224 |
| [43] | Douglas, M., Lu, Z.: On the geometry of moduli space of polarized Calabi-Yau manifolds. arXiv:math/0603414 [math-dg] |
| [44] | Giveon, A., Kutasov, D., Seiberg, N.: Comments on string theory on AdS(3). Adv. Theor. Math. Phys. 2, 733 (1998). arXiv:hep-th/9806194 · Zbl 1041.81575 |
| [45] | Seiberg, N., Witten, E.: The D1/D5 system and singular CFT. JHEP 9904, 017 (1999). arXiv:hep-th/9903224 · Zbl 0953.81076 |
| [46] | Cheng, M.C.N., Duncan, J.F.R.: Rademacher sums and rademacher series. arXiv:1210.3066 [math.NT] · Zbl 1344.11037 |
| [47] | Galakhov, D., Longhi, P., Mainiero, T., Moore, G.W., Neitzke, A.: Wild wall crossing and BPS giants. JHEP 1311, 046 (2013). arXiv:1305.5454 [hep-th] |
| [48] | Friedan, D.: A Tentative theory of large distance physics. JHEP 0310, 063 (2003). arXiv:hep-th/0204131 |
| [49] | Friedan, D.: Two talks on a tentative theory of large distance physics. arXiv:hep-th/0212268 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
