Abstract
When the SU(N) \( \mathcal{N} \) = 4 super-Yang-Mills (SYM) theory with complexified gauge coupling τ is placed on a round four-sphere and deformed by an \( \mathcal{N} \) = 2-preserving mass parameter m, its free energy F (m, τ, \( \overline{\tau} \)) can be computed exactly using supersymmetric localization. In this work, we derive a new exact relation between the fourth derivative \( {\partial}_m^4F\left(m,\tau, \overline{\tau}\right)\left|{{}_m}_{=0}\right. \) of the sphere free energy and the integrated stress-tensor multiplet four-point function in the \( \mathcal{N} \) = 4 SYM theory. We then apply this exact relation, along with various other constraints derived in previous work (coming from analytic bootstrap, the mixed derivative \( {\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2F\left(m,\tau, \overline{\tau}\right)\left|{{}_m}_{=0}\right. \), and type IIB superstring theory scattering amplitudes) to determine various perturbative terms in the large N and large ’t Hooft coupling λ expansion of the \( \mathcal{N} \) = 4 SYM correlator at separated points. In particular, we determine the leading large-λ term in the \( \mathcal{N} \) = 4 SYM correlation function at order 1/N8. This is three orders beyond the planar limit.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F. Gonzalez-Rey, I.Y. Park and K. Schalm, A note on four point functions of conformal operators in N = 4 super-Yang-Mills, Phys. Lett. B 448 (1999) 37 [hep-th/9811155] [INSPIRE].
B. Eden, C. Schubert and E. Sokatchev, Three loop four point correlator in N = 4 SYM, Phys. Lett. B 482 (2000) 309 [hep-th/0003096] [INSPIRE].
M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Anomalous dimensions in N = 4 SYM theory at order g4, Nucl. Phys. B 584 (2000) 216 [hep-th/0003203] [INSPIRE].
J. Drummond, C. Duhr, B. Eden, P. Heslop, J. Pennington and V.A. Smirnov, Leading singularities and off-shell conformal integrals, JHEP 08 (2013) 133 [arXiv:1303.6909] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [Adv. Theor. Math. Phys. 2 (1998) 231] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
B. Eden, A.C. Petkou, C. Schubert and E. Sokatchev, Partial nonrenormalization of the stress tensor four point function in N = 4 SYM and AdS/CFT, Nucl. Phys. B 607 (2001) 191 [hep-th/0009106] [INSPIRE].
G. Arutyunov, F.A. Dolan, H. Osborn and E. Sokatchev, Correlation functions and massive Kaluza-Klein modes in the AdS/CFT correspondence, Nucl. Phys. B 665 (2003) 273 [hep-th/0212116] [INSPIRE].
G. Arutyunov and E. Sokatchev, On a large N degeneracy in N = 4 SYM and the AdS/CFT correspondence, Nucl. Phys. B 663 (2003) 163 [hep-th/0301058] [INSPIRE].
L. Berdichevsky and P. Naaijkens, Four-point functions of different-weight operators in the AdS/CFT correspondence, JHEP 01 (2008) 071 [arXiv:0709.1365] [INSPIRE].
L.I. Uruchurtu, Four-point correlators with higher weight superconformal primaries in the AdS/CFT Correspondence, JHEP 03 (2009) 133 [arXiv:0811.2320] [INSPIRE].
L.I. Uruchurtu, Next-next-to-extremal four point functions of N = 4 1/2 BPS operators in the AdS/CFT correspondence, JHEP 08 (2011) 133 [arXiv:1106.0630] [INSPIRE].
G. Arutyunov, S. Frolov, R. Klabbers and S. Savin, Towards 4-point correlation functions of any \( \frac{1}{2} \)-BPS operators from supergravity, JHEP 04 (2017) 005 [arXiv:1701.00998] [INSPIRE].
G. Arutyunov, R. Klabbers and S. Savin, Four-point functions of 1/2-BPS operators of any weights in the supergravity approximation, JHEP 09 (2018) 118 [arXiv:1808.06788] [INSPIRE].
S. de Haro, A. Sinkovics and K. Skenderis, On a supersymmetric completion of the R4 term in 2B supergravity, Phys. Rev. D 67 (2003) 084010 [hep-th/0210080] [INSPIRE].
G. Policastro and D. Tsimpis, R4 purified, Class. Quant. Grav. 23 (2006) 4753 [hep-th/0603165] [INSPIRE].
M.F. Paulos, Higher derivative terms including the Ramond-Ramond five-form, JHEP 10 (2008) 047 [arXiv:0804.0763] [INSPIRE].
J.T. Liu and R. Minasian, Higher-derivative couplings in string theory: dualities and the B-field, Nucl. Phys. B 874 (2013) 413 [arXiv:1304.3137] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
L. Rastelli and X. Zhou, How to succeed at holographic correlators without really trying, JHEP 04 (2018) 014 [arXiv:1710.05923] [INSPIRE].
S. Caron-Huot and A.-K. Trinh, All tree-level correlators in AdS5S5 supergravity: hidden ten-dimensional conformal symmetry, JHEP 01 (2019) 196 [arXiv:1809.09173] [INSPIRE].
V. Gonçalves, R. Pereira and X. Zhou, 20′ five-point function from AdS5 × S5 supergravity, JHEP 10 (2019) 247 [arXiv:1906.05305] [INSPIRE].
L. Rastelli and X. Zhou, Holographic four-point functions in the (2, 0) theory, JHEP 06 (2018) 087 [arXiv:1712.02788] [INSPIRE].
L. Rastelli, K. Roumpedakis and X. Zhou, AdS3 × S3 tree-level correlators: hidden six-dimensional conformal symmetry, JHEP 10 (2019) 140 [arXiv:1905.11983] [INSPIRE].
X. Zhou, On superconformal four-point Mellin amplitudes in dimension d > 2, JHEP 08 (2018) 187 [arXiv:1712.02800] [INSPIRE].
X. Zhou, On Mellin amplitudes in SCFTs with eight supercharges, JHEP 07 (2018) 147 [arXiv:1804.02397] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Lessons from crossing symmetry at large N, JHEP 06 (2015) 074 [arXiv:1410.4717] [INSPIRE].
S.M. Chester, AdS4/CFT3 for unprotected operators, JHEP 07 (2018) 030 [arXiv:1803.01379] [INSPIRE].
S.M. Chester, S.S. Pufu and X. Yin, The M-theory S-matrix from ABJM: beyond 11D supergravity, JHEP 08 (2018) 115 [arXiv:1804.00949] [INSPIRE].
S.M. Chester and E. Perlmutter, M-theory reconstruction from (2, 0) CFT and the chiral algebra conjecture, JHEP 08 (2018) 116 [arXiv:1805.00892] [INSPIRE].
D.J. Binder, S.M. Chester and S.S. Pufu, Absence of D4R4 in M-theory From ABJM, JHEP 04 (2020) 052 [arXiv:1808.10554] [INSPIRE].
D.J. Binder, S.M. Chester and S.S. Pufu, AdS4/CFT3 from weak to strong string coupling, JHEP 01 (2020) 034 [arXiv:1906.07195] [INSPIRE].
F. Aprile, J. Drummond, P. Heslop and H. Paul, Double-trace spectrum of N = 4 supersymmetric Yang-Mills theory at strong coupling, Phys. Rev. D 98 (2018) 126008 [arXiv:1802.06889] [INSPIRE].
S. Giusto, R. Russo and C. Wen, Holographic correlators in AdS3, JHEP 03 (2019) 096 [arXiv:1812.06479] [INSPIRE].
S. Giusto, R. Russo, A. Tyukov and C. Wen, Holographic correlators in AdS3 without Witten diagrams, JHEP 09 (2019) 030 [arXiv:1905.12314] [INSPIRE].
L.F. Alday, A. Bissi and E. Perlmutter, Holographic reconstruction of AdS exchanges from crossing symmetry, JHEP 08 (2017) 147 [arXiv:1705.02318] [INSPIRE].
L.F. Alday and E. Perlmutter, Growing extra dimensions in AdS/CFT, JHEP 08 (2019) 084 [arXiv:1906.01477] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
D.J. Binder, S.M. Chester, S.S. Pufu and Y. Wang, \( \mathcal{N} \) = 4 Super-Yang-Mills correlators at strong coupling from string theory and localization, JHEP 12 (2019) 119 [arXiv:1902.06263] [INSPIRE].
S.M. Chester, Genus-2 holographic correlator on AdS5 × S5 from localization, JHEP 04 (2020) 193 [arXiv:1908.05247] [INSPIRE].
S.M. Chester, M.B. Green, S.S. Pufu, Y. Wang and C. Wen, Modular invariance in superstring theory from \( \mathcal{N} \) = 4 super-Yang-Mills, JHEP 11 (2020) 016 [arXiv:1912.13365] [INSPIRE].
J. Polchinski, S matrices from AdS space-time, hep-th/9901076 [INSPIRE].
L. Susskind, Holography in the flat space limit, AIP Conf. Proc. 493 (1999) 98 [hep-th/9901079] [INSPIRE].
S.B. Giddings, Flat space scattering and bulk locality in the AdS/CFT correspondence, Phys. Rev. D 61 (2000) 106008 [hep-th/9907129] [INSPIRE].
A. Fitzpatrick and J. Kaplan, Analyticity and the Holographic S-matrix, JHEP 10 (2012) 127 [arXiv:1111.6972] [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
A. Fitzpatrick and J. Kaplan, Scattering states in AdS/CFT, arXiv:1104.2597 [INSPIRE].
V. Gonçalves, Four point function of \( \mathcal{N} \) = 4 stress-tensor multiplet at strong coupling, JHEP 04 (2015) 150 [arXiv:1411.1675] [INSPIRE].
O. Aharony, L.F. Alday, A. Bissi and E. Perlmutter, Loops in AdS from conformal field theory, JHEP 07 (2017) 036 [arXiv:1612.03891] [INSPIRE].
L.F. Alday and A. Bissi, Loop corrections to supergravity on AdS5 × S5, Phys. Rev. Lett. 119 (2017) 171601 [arXiv:1706.02388] [INSPIRE].
F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Quantum gravity from conformal field theory, JHEP 01 (2018) 035 [arXiv:1706.02822] [INSPIRE].
F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Unmixing supergravity, JHEP 02 (2018) 133 [arXiv:1706.08456] [INSPIRE].
F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Loop corrections for Kaluza-Klein AdS amplitudes, JHEP 05 (2018) 056 [arXiv:1711.03903] [INSPIRE].
L.F. Alday, A. Bissi and E. Perlmutter, Genus-one string amplitudes from conformal field theory, JHEP 06 (2019) 010 [arXiv:1809.10670] [INSPIRE].
L.F. Alday, On genus-one string amplitudes on AdS5 × S5, arXiv:1812.11783 [INSPIRE].
J.M. Drummond, D. Nandan, H. Paul and K.S. Rigatos, String corrections to AdS amplitudes and the double-trace spectrum of \( \mathcal{N} \) = 4 SYM, JHEP 12 (2019) 173 [arXiv:1907.00992] [INSPIRE].
L.F. Alday and S. Caron-Huot, Gravitational S-matrix from CFT dispersion relations, JHEP 12 (2018) 017 [arXiv:1711.02031] [INSPIRE].
J.M. Drummond and H. Paul, One-loop string corrections to AdS amplitudes from CFT, arXiv:1912.07632 [INSPIRE].
F. Aprile, J. Drummond, P. Heslop and H. Paul, One-loop amplitudes in AdS5 × S5 supergravity from \( \mathcal{N} \) = 4 SYM at strong coupling, JHEP 03 (2020) 190 [arXiv:1912.01047] [INSPIRE].
L.F. Alday and X. Zhou, Simplicity of AdS supergravity at one loop, JHEP 09 (2020) 008 [arXiv:1912.02663] [INSPIRE].
F.A. Dolan and H. Osborn, Superconformal symmetry, correlation functions and the operator product expansion, Nucl. Phys. B 629 (2002) 3 [hep-th/0112251] [INSPIRE].
E. Gerchkovitz, J. Gomis, N. Ishtiaque, A. Karasik, Z. Komargodski and S.S. Pufu, Correlation functions of coulomb branch operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].
J. Gomis and N. Ishtiaque, Kähler potential and ambiguities in 4d \( \mathcal{N} \) = 2 SCFTs, JHEP 04 (2015) 169 [arXiv:1409.5325] [INSPIRE].
E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere partition functions and the Zamolodchikov metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
A.V. Belitsky, S. Hohenegger, G.P. Korchemsky and E. Sokatchev, N = 4 superconformal Ward identities for correlation functions, Nucl. Phys. B 904 (2016) 176 [arXiv:1409.2502] [INSPIRE].
B. Eynard, Topological expansion for the 1-Hermitian matrix model correlation functions, JHEP 11 (2004) 031 [hep-th/0407261] [INSPIRE].
B. Eynard and N. Orantin, Algebraic methods in random matrices and enumerative geometry, arXiv:0811.3531 [INSPIRE].
M.L. Mehta, A method of integration over matrix variables, Comm. Math. Phys. 79 (1981) 327.
A. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A natural language for AdS/CFT correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].
A. Fitzpatrick and J. Kaplan, Unitarity and the holographic S-matrix, JHEP 10 (2012) 032 [arXiv:1112.4845] [INSPIRE].
J. Penedones, J.A. Silva and A. Zhiboedov, Nonperturbative Mellin amplitudes: existence, properties, applications, JHEP 08 (2020) 031 [arXiv:1912.11100] [INSPIRE].
G. Arutyunov and S. Frolov, Four point functions of lowest weight CPOs in N = 4 SYM(4) in supergravity approximation, Phys. Rev. D 62 (2000) 064016 [hep-th/0002170] [INSPIRE].
H. Gomez and C.R. Mafra, The overall coefficient of the two-loop superstring amplitude using pure spinors, JHEP 05 (2010) 017 [arXiv:1003.0678] [INSPIRE].
E. D’Hoker, M. Gutperle and D.H. Phong, Two-loop superstrings and S-duality, Nucl. Phys. B 722 (2005) 81 [hep-th/0503180] [INSPIRE].
H. Gomez and C.R. Mafra, The closed-string 3-loop amplitude and S-duality, JHEP 10 (2013) 217 [arXiv:1308.6567] [INSPIRE].
M.B. Green, J.G. Russo and P. Vanhove, Low energy expansion of the four-particle genus-one amplitude in type-II superstring theory, JHEP 02 (2008) 020 [arXiv:0801.0322] [INSPIRE].
J.G. Russo and K. Zarembo, Massive N = 2 gauge theories at large N, JHEP 11 (2013) 130 [arXiv:1309.1004] [INSPIRE].
G. Arutyunov, S. Frolov and A.C. Petkou, Operator product expansion of the lowest weight CPOs in \( \mathcal{N} \) = 4 SYM4 at strong coupling, Nucl. Phys. B 586 (2000) 547 [Erratum ibid. 609 (2001) 539] [hep-th/0005182] [INSPIRE].
E. D’Hoker, S.D. Mathur, A. Matusis and L. Rastelli, The operator product expansion of N = 4 SYM and the 4 point functions of supergravity, Nucl. Phys. B 589 (2000) 38 [hep-th/9911222] [INSPIRE].
M.B. Green, M. Gutperle and P. Vanhove, One loop in eleven-dimensions, Phys. Lett. B 409 (1997) 177 [hep-th/9706175] [INSPIRE].
M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D 59 (1999) 046006 [hep-th/9808061] [INSPIRE].
M.B. Green, H.-h. Kwon and P. Vanhove, Two loops in eleven-dimensions, Phys. Rev. D 61 (2000) 104010 [hep-th/9910055] [INSPIRE].
M.B. Green and P. Vanhove, Duality and higher derivative terms in M-theory, JHEP 01 (2006) 093 [hep-th/0510027] [INSPIRE].
M.B. Green, S.D. Miller and P. Vanhove, SL(2, ℤ)-invariance and D-instanton contributions to the D6R4 interaction, Commun. Num. Theor. Phys. 09 (2015) 307 [arXiv:1404.2192] [INSPIRE].
S. Chester, M. Green, S. Pufu, Y. Wang and C. Wen, New modular invariants in N = 4 super-Yang-Mills theory, arXiv:2008.02713.
N. Hama and K. Hosomichi, Seiberg-Witten theories on ellipsoids, JHEP 09 (2012) 033 [Addendum ibid. 10 (2012) 051] [arXiv:1206.6359] [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, L. Rastelli and B.C. van Rees, More \( \mathcal{N} \) = 4 superconformal bootstrap, Phys. Rev. D 96 (2017) 046014 [arXiv:1612.02363] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
A. Losev, N. Nekrasov and S.L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B 534 (1998) 549 [hep-th/9711108] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2003.08412
Supplementary Information
ESM 1
(NB 2335 kb)
Rights and permissions
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.
About this article
Cite this article
Chester, S.M., Pufu, S.S. Far beyond the planar limit in strongly-coupled \( \mathcal{N} \) = 4 SYM. J. High Energ. Phys. 2021, 103 (2021). https://doi.org/10.1007/JHEP01(2021)103
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2021)103