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  • Cited by 2
  • Print publication year: 2012
  • Online publication date: August 2012

9 - Emergent spacetime


We give an introductory account of the AdS/CFT correspondence in the ½-BPS sector of N =4 super Yang-Mills theory. Six of the dimensions of the string theory are emergent in the Yang-Mills theory. In this chapter we suggest how these dimensions and local physics in these dimensions emerge. The discussion is aimed at non-experts.


The problem of quantizing gravity has proved to be a difficult one. To solve this problem, it seems to be necessary to answer the question “What is spacetime?” This challenges the most basic assumptions we are used to making; a radical new idea may be needed. Further, the hope of any guidance from experiment seems to be out of the question. One might conclude that the situation is hopeless. Drawing on recent insights from the AdS/CFT correspondence, we are nonetheless, optimistic.

The AdS/CFT correspondence [1] claims an exact equality between N =4 super Yang-Mills theory in flat (3+1)-dimensional Minkowski spacetime and Type IIB string theory on an asymptotically AdS5×S5 background. Type IIB string theory is a theory of closed strings; at least within string perturbation theory, theories of closed strings provide a consistent UV completion of gravity. The fact that such an equality exists is highly unexpected and non-trivial, and (as we will try to convince the reader) can be used to gain insight into the nature of spacetime. George Ellis opened the Foundations of Space and Time workshop by holding up two fingers and asking “are there an infinite or a finite number of places a particle could occupy between my fingers?” We don't know the answer to George's question.

[1] J. M., Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200];
S. S., Gubser, I. R., Klebanov, and A. M., Polyakov, “Gauge theory correlators from non-critical string theory,” Phys. Lett.B 428, 105 (1998) [arXiv:hep-th/9802109];
E., Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150].
[2] S. R., Das and A., Jevicki, “String field theory and physical interpretation of D = 1 Strings,” Mod. Phys. Lett.A 5, 1639 (1990).
[3] S., Corley, A., Jevicki, and S., Ramgoolam, “Exact correlators of giant gravitons from dual N = 4 SYM theory,” Adv. Theor. Math. Phys. 5, 809 (2002) [arXiv:hep-th/0111222].
[4] D., Berenstein, “A toy model for the AdS]CFT correspondence,” JHEP 0407, 018 (2004) [arXiv:hep-th/0403110].
[5] H., Lin, O., Lunin, and J.M., Maldacena, “Bubbling AdS space and 1/2 BPS geometries,” JHEP 0410, 025 (2004) [arXiv:hep-th/0409174].
[6] V., Balasubramanian, J., de Boer, V., Jejjala, and J., Simon, “The library of Babel: On the origin of gravitational thermodynamics,” JHEP 0512, 006 (2005) [arXiv:hep-th/0508023];
V., Balasubramanian, V., Jejjala, and J., Simon, “The library of Babel,” Int. J. Mod. Phys.D 14, 2181 (2005) [arXiv:hep-th/0505123].
[7] D., Berenstein, “Large N BPS states and emergent quantum gravity,” JHEP 0601, 125 (2006) [arXiv:hep-th/0507203].
[8] V., Balasubramanian, D., Berenstein, B., Feng, and M. x., Huang, “D-branes in Yang–Mills theory and emergent gauge symmetry,” JHEP 0503, 006 (2005) [arXiv:hep-th/0411205].
[9] S., Ramgoolam, “Schur–Weyl duality as an instrument of gauge–string duality,” arXiv:0804.2764 [hep-th].
[10] D., Berenstein, J. M., Maldacena, and H., Nastase, “Strings in flat space and pp waves from N = 4 super Yang–Mills,” JHEP 0204, 013 (2002) [arXiv:hep-th/0202021].
[11] E., Brezin, C., Itzykson, G., Parisi, and J. B., Zuber, “Planar diagrams,” Commun. Math. Phys. 59, 35 (1978).
[12] R. C., Myers, “Dielectric-branes,” JHEP 9912, 022 (1999) [arXiv:hep-th/9910053].
[13] J., McGreevy, L., Susskind, and N., Toumbas, “Invasion of the giant gravitons from anti-de Sitter space,” JHEP 0006, 008 (2000) [arXiv:hep-th/0003075].
[14] V., Balasubramanian, M., Berkooz, A., Naqvi, and M. J., Strassler, “Giant gravitons in conformal field theory,” JHEP 0204, 034 (2002) [arXiv:hep-th/0107119].
[15] R., de Mello Koch and R., Gwyn, “Giant graviton correlators from dual SU(N) super Yang–Mills theory,” JHEP 0411, 081 (2004) [arXiv:hep-th/0410236];
T. W., Brown, “Half-BPS SU(N) correlators in N = 4 SYM,” arXiv:hep-th/0703202.
[16] T. W., Brown, R., de Mello Koch, S., Ramgoolam, and N., Toumbas, “Correlators, probabilities and topologies in N = 4 SYM,” JHEP 0703, 072 (2007) [arXiv:hep-th/0611290].
[17] S., Corley and S., Ramgoolam, “Finite factorization equations and sumrules for BPS correlators in N = 4 SYM theory,” Nucl. Phys.B 641, 131 (2002) [arXiv:hep-th/0205221].
[18] T. W., Brown, P. J., Heslop, and S., Ramgoolam, “Diagonal multi-matrix correlators and BPS operators in N = 4 SYM,” arXiv:0711.0176 [hep-th];
T. W., Brown, “Permutations and the loop,” JHEP 0806, 008 (2008) [arXiv:0801.2094 [hep-th]];
T.W., Brown, P. J., Heslop, and S., Ramgoolam, “Diagonal free field matrix correlators, global symmetries and giant gravitons,” JHEP 0904, 089 (2009) [arXiv:0806.1911 [hep-th]].
[19] Y., Kimura and S., Ramgoolam, “Branes, anti-branes and Brauer algebras in gauge–gravity duality,” arXiv:0709.2158 [hep-th];
Y., Kimura, “Non-holomorphic multi-matrix gauge invariant operators based on Brauer algebra,” arXiv:0910.2170 [hep-th].
[20] R., Bhattacharyya, S., Collins, and R., de Mello Koch, “Exact multi-matrix correlators,” arXiv:0801.2061 [hep-th];
R., Bhattacharyya, R., de Mello Koch, and M., Stephanou, “Exact multi-restricted Schur polynomial correlators,” JHEP 0806, 101 (2008) [arXiv:0805.3025 [hep-th]];
S., Collins, “Restricted Schur polynomials and finite N counting,” Phys. Rev.D 79, 026002 (2009) [arXiv:0810.4217 [hep-th]].
[21] Y., Kimura and S., Ramgoolam, “Enhanced symmetries of gauge theory and resolving the spectrum of local operators,” Phys. Rev.D 78, 126003 (2008) [arXiv:0807.3696 [hep-th]].
[22] M., Bianchi, D. Z., Freedman, and K., Skenderis, “How to go with an RG flow,” JHEP 0108, 041 (2001) [arXiv:hep-th/0105276];
M., Bianchi, D. Z., Freedman, and K., Skenderis, “Holographic renormalization,” Nucl. Phys.B 631, 159 (2002) [arXiv:hep-th/0112119];
K., Skenderis, “Lecture notes on holographic renormalization,” Class. Quant. Grav. 19, 5849 (2002) [arXiv:hep-th/0209067].
[23] K., Skenderis and M., Taylor, “Kaluza–Klein holography,” JHEP 0605, 057 (2006) [arXiv:hep-th/0603016];
K., Skenderis and M., Taylor, “Anatomy of bubbling solutions,” JHEP 0709, 019 (2007) [arXiv:0706.0216 [hep-th]].
[24] S., Lee, S., Minwalla, M., Rangamani, and N., Seiberg, “Three-point functions of chiral operators in D =4, N = 4 SYM at large N,” Adv. Theor. Math. Phys. 2, 697 (1998) [arXiv:hep-th/9806074];
K. A., Intriligator, “Bonus symmetries of N = 4 super-Yang–Mills correlation functions via AdS duality,” Nucl. Phys.B 551, 575 (1999) [arXiv:hep-th/9811047];
B. U., Eden, P. S., Howe, A., Pickering, E., Sokatchev and P. C., West, “Four-point functions in N = 2 superconformal field theories,” Nucl. Phys.B 581, 523 (2000) [arXiv:hep-th/0001138];
B. U., Eden, P. S., Howe, E., Sokatchev and P. C., West, “Extremal and next-to-extremal n-point correlators in four-dimensional SCFT,” Phys. Lett.B 494, 141 (2000) [arXiv:hep-th/0004102].
[25] C., Kristjansen, J., Plefka, G. W., Semenoff and M., Staudacher, “A new double-scaling limit of N = 4 super Yang–Mills theory and PP-wave strings,” Nucl. Phys.B 643, 3 (2002) [arXiv:hep-th/0205033];
N. R., Constable, D. Z., Freedman, M., Headrick, S., Minwalla, L., Motl, A., Postnikov and W., Skiba, “PP-wave string interactions from perturbative Yang–Mills theory,” JHEP 0207, 017 (2002) [arXiv:hep-th/0205089].
[26] J.A., Minahan and K., Zarembo, “The Bethe-ansatz for N =4 super Yang–Mills,” JHEP 0303, 013 (2003) [arXiv:hep-th/0212208].
[27] N., Beisert and M., Staudacher, “The N =4 SYM Integrable Super Spin Chain,” Nucl. Phys.B 670, 439 (2003) [arXiv:hep-th/0307042].
[28] N., Beisert, C., Kristjansen and M., Staudacher, “The dilatation operator of N =4 super Yang–Mills theory,” Nucl. Phys.B 664, 131 (2003) [arXiv:hep-th/0303060].
[29] M., Kruczenski, “Spin chains and string theory,” Phys. Rev. Lett. 93, 161602 (2004) [arXiv:hep-th/0311203].
M., Kruczenski, A. V., Ryzhov and A.A., Tseytlin, “Large spin limit of AdS(5) × S**5 string theory and low energy expansion of ferromagnetic spin chains,” Nucl. Phys.B 692, 3 (2004) [arXiv:hep-th/0403120].
[30] M. T., Grisaru, R. C., Myers and O., Tafjord, “SUSY and Goliath,” JHEP 0008, 040 (2000) [arXiv:hep-th/0008015].
[31] A., Hashimoto, S., Hirano and N., Itzhaki, “Large branes in AdS and their field theory dual,” JHEP 0008, 051 (2000) [arXiv:hep-th/0008016].
[32] R., de Mello Koch, J., Smolic and M., Smolic, “Giant Gravitons - with Strings Attached (I),” JHEP 0706, 074 (2007), arXiv:hep-th/0701066.
[33] D., Berenstein, D. H., Correa and S. E., Vazquez, “A study of open strings ending on giant gravitons, spin chains and integrability,” [arXiv:hep-th/0604123];
D., Berenstein, D. H., Correa and S. E., Vazquez, “Quantizing open spin chains with variable length: An example from giant gravitons,” Phys. Rev. Lett. 95, 191601 (2005) [arXiv:hep-th/0502172];
D. H., Correa and G. A., Silva, “Dilatation operator and the super Yang–Mills duals of open strings on AdS giant gravitons,” JHEP 0611, 059 (2006) [arXiv:hep-th/0608128].
[34] R., de Mello Koch, J., Smolic and M., Smolic, “Giant gravitons – with strings attached (II),” JHEP 0709 049 (2007) [arXiv:hep-th/0701067];
D., Bekker, R., de Mello Koch and M., Stephanou, “Giant gravitons – with strings attached (III),” JHEP 0802, 029 (2008) [arXiv:0710.5372 [hep-th]].
[35] A., Hamilton and J., Murugan, “On the shoulders of giants – quantum gravity and braneworld stability,” [arXiv:0806.3273 [gr-qc]]
[36] L., Grant, L., Maoz, J., Marsano, K., Papadodimas and V. S., Rychkov, “Minisuperspace quantization of ‘bubbling AdS’ and free fermion droplets,” JHEP 0508, 025 (2005) [arXiv:hep-th/0505079];
L., Maoz and V. S., Rychkov, “Geometry quantization from supergravity: The case of ‘bubbling AdS’,” JHEP 0508, 096 (2005) [arXiv:hep-th/0508059].
[37] R., de Mello Koch, “Geometries from Young diagrams,” JHEP 0811, 061 (2008) [arXiv:0806.0685 [hep-th]].
[38] R., de Mello Koch, N., Ives, and M., Stephanou, “Correlators in nontrivial backgrounds,” Phys. Rev.D 79, 026004 (2009) [arXiv:0810.4041 [hep-th]].
[39] K., Skenderis and M., Taylor, “Anatomy of bubbling solutions,” JHEP 0709, 019 (2007) [arXiv:0706.0216 [hep-th]].
[40] R., de Mello Koch, T. K., Dey, N., Ives, and M., Stephanou, “Correlators of operators with a large R-charge,” arXiv:0905.2273 [hep-th].
[41] S. E., Vazquez, “Reconstructing 1/2 BPS space-time metrics from matrix models and spin chains,” Phys. Rev.D 75, 125012 (2007) [arXiv:hep-th/0612014].
[42] H.Y., Chen, D. H., Correa, and G.A., Silva, “Geometry and topology of bubble solutions from gauge theory,” Phys. Rev.D 76, 026003 (2007) [arXiv:hep-th/0703068].
[43] G., Mandal, “Fermions from half-BPS supergravity,” JHEP 0508, 052 (2005) [arXiv:hep-th/0502104].
[44] M., Masuku and J. P., Rodrigues, “Laplacians in polar matrix coordinates and radial fermionization in higher dimensions,” arXiv:0911.2846 [hep-th];
Y., Kimura, S., Ramgoolam, and D., Turton, “Free particles from Brauer algebras in complexmatrix models,” arXiv:0911.4408 [hep-th].
[45] D., Berenstein, “A strong coupling expansion for N = 4 SYM theory and other SCFT's,” arXiv:0804.0383 [hep-th];
D. E., Berenstein and S. A., Hartnoll, “Strings on conifolds from strong coupling dynamics: quantitative results,” JHEP 0803 (2008) 072 [arXiv:0711.3026 [hep-th]];
D., Berenstein, “Strings on conifolds from strong coupling dynamics, part I,” JHEP 0804 (2008) 002 [arXiv:0710.2086 [hep-th]];
D. E., Berenstein, M., Hanada, and S.A., Hartnoll, “Multi-matrix models and emergent geometry,” JHEP 0902, 010 (2009) [arXiv:0805.4658 [hep-th]].
[46] A., Jevicki and B., Sakita, “The quantum collective field method and its application to the planar limit,” Nucl. Phys.B 165, 511 (1980);
A., Jevicki and B., Sakita, “Collective field approach to the large N limit: Euclidean field theories,” Nucl. Phys.B 185, 89 (1981).
[47] J. P., Rodrigues, “Large N spectrum of two matrices in a harmonic potential and BMN energies,” JHEP 0512, 043 (2005) [arXiv:hep-th/0510244];
A., Donos, A., Jevicki, and J. P., Rodrigues, “Matrix model maps in AdS/CFT,” Phys. Rev.D 72, 125009 (2005) [arXiv:hep-th/0507124];
R., de Mello Koch, A., Jevicki, and J. P., Rodrigues, “Instantons in c=0 CSFT,” JHEP 0504, 011 (2005) [arXiv:hep-th/0412319];
R., de Mello Koch, A., Donos, A., Jevicki, and J. P., Rodrigues, “Derivation of string field theory from the large N BMN limit,” Phys. Rev.D 68, 065012 (2003) [arXiv:hep-th/0305042];
R., de Mello Koch, A., Jevicki, and J. P., Rodrigues, “Collective string field theory of matrix models in the BMN limit,” Int. J. Mod. Phys.A 19, 1747 (2004) [arXiv:hep-th/0209155].
[48] A., Donos, “A description of 1/4 BPS configurations in minimal type IIB SUGRA,” Phys. Rev.D 75, 025010 (2007) [arXiv:hep-th/0606199/;
B., Chen et al., “Bubbling AdS and droplet descriptions of BPS geometries in IIB supergravity,” JHEP 0710, 003 (2007) [arXiv:0704.2233 [hep-th]];
O., Lunin, “Brane webs and 1/4-BPS geometries,” arXiv:0802.0735 [hep-th].