Abstract
We consider the integrated correlators associated with four-point correlation functions \( \left\langle {\mathcal{O}}_2{\mathcal{O}}_2{\mathcal{O}}_p^{(i)}{\mathcal{O}}_p^{(j)}\right\rangle \) in four-dimensional \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory (SYM) with SU(N) gauge group, where \( {\mathcal{O}}_p^{(i)} \) is a superconformal primary with charge (or dimension) p and the superscript i represents possible degeneracy. These integrated correlators are defined by integrating out spacetime dependence with a certain integration measure, and they can be computed via supersymmetric localisation. They are modular functions of complexified Yang-Mills coupling τ. We show that the localisation computation is systematised by appropriately reorganising the operators. After this reorganisation of the operators, we prove that all the integrated correlators for any N, with some crucial normalisation factor, satisfy a universal Laplace-difference equation (with the laplacian defined on the τ-plane) that relates integrated correlators of operators with different charges. This Laplace-difference equation is a recursion relation that completely determines all the integrated correlators, once the initial conditions are given.
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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 and S.S. Pufu, Far beyond the planar limit in strongly-coupled \( \mathcal{N} \) = 4 SYM, JHEP 01 (2021) 103 [arXiv:2003.08412] [INSPIRE].
S.M. Chester, Genus-2 holographic correlator on AdS5 × S5 from localization, JHEP 04 (2020) 193 [arXiv:1908.05247] [INSPIRE].
S.M. Chester et al., Modular invariance in superstring theory from \( \mathcal{N} \) = 4 super-Yang-Mills, JHEP 11 (2020) 016 [arXiv:1912.13365] [INSPIRE].
S.M. Chester et al., New modular invariants in \( \mathcal{N} \) = 4 Super-Yang-Mills theory, JHEP 04 (2021) 212 [arXiv:2008.02713] [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].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
D. Dorigoni, M.B. Green and C. Wen, Novel Representation of an Integrated Correlator in \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 126 (2021) 161601 arXiv:2102.08305] [INSPIRE].
D. Dorigoni, M.B. Green and C. Wen, Exact properties of an integrated correlator in \( \mathcal{N} \) = 4 SU(N) SYM, JHEP 05 (2021) 089 [arXiv:2102.09537] [INSPIRE].
D. Dorigoni, M.B. Green and C. Wen, Exact results for duality-covariant integrated correlators in \( \mathcal{N} \) = 4 SYM with general classical gauge groups, SciPost Phys. 13 (2022) 092 [arXiv:2202.05784] [INSPIRE].
D. Dorigoni, M.B. Green and C. Wen, The SAGEX review on scattering amplitudes Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their supersymmetric Yang-Mills duals, J. Phys. A 55 (2022) 443011 [arXiv:2203.13021] [INSPIRE].
H. Paul, E. Perlmutter and H. Raj, Integrated Correlators in \( \mathcal{N} \) = 4 SYM via SL(2, ℤ) Spectral Theory, JHEP 01 (2023) 149 [arXiv:2209.06639] [INSPIRE].
C. Rayson, Some aspects of conformal \( \mathcal{N} \) = 4 SYM four point function, Ph.D. thesis, Cambridge University, Cambridge. U.K. (2008) [arXiv:1706.04450] [INSPIRE].
F. Aprile et al., Single particle operators and their correlators in free \( \mathcal{N} \) = 4 SYM, JHEP 11 (2020) 072 [arXiv:2007.09395] [INSPIRE].
E. D’Hoker, J. Erdmenger, D.Z. Freedman and M. Perez-Victoria, Near extremal correlators and vanishing supergravity couplings in AdS/CFT, Nucl. Phys. B 589 (2000) 3 [hep-th/0003218].
B. Fiol and Z. Kong, The planar limit of integrated 4-point functions, arXiv:2303.09572 [INSPIRE].
C. Montonen and D.I. Olive, Magnetic Monopoles as Gauge Particles?, Phys. Lett. B 72 (1977) 117 [INSPIRE].
E. Gerchkovitz et al., Correlation Functions of Coulomb Branch Operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].
H. Paul, E. Perlmutter and H. Raj, Exact Large Charge in \( \mathcal{N} \) = 4 SYM and Semiclassical String Theory, arXiv:2303.13207 [INSPIRE].
M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [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].
M. Nirschl and H. Osborn, Superconformal Ward identities and their solution, Nucl. Phys. B 711 (2005) 409 [hep-th/0407060] [INSPIRE].
F. Fucito, J.F. Morales and R. Poghossian, Wilson loops and chiral correlators on squashed spheres, JHEP 11 (2015) 064 [arXiv:1507.05426] [INSPIRE].
J.G. Russo and K. Zarembo, Massive N = 2 Gauge Theories at Large N, JHEP 11 (2013) 130 [arXiv:1309.1004] [INSPIRE].
M.B. Green and C. Wen, Maximal U(1)Y-violating n-point correlators in \( \mathcal{N} \) = 4 super-Yang-Mills theory, JHEP 02 (2021) 042 [arXiv:2009.01211] [INSPIRE].
D. Dorigoni, M.B. Green and C. Wen, Exact expressions for n-point maximal U(1)Y-violating integrated correlators in SU(N) \( \mathcal{N} \) = 4 SYM, JHEP 11 (2021) 132 [arXiv:2109.08086] [INSPIRE].
D. Dorigoni, M.B. Green, C. Wen and H. Xie, Modular-invariant large-N completion of an integrated correlator in \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory, JHEP 04 (2023) 114 [arXiv:2210.14038] [INSPIRE].
L.F. Alday, S.M. Chester and T. Hansen, Modular invariant holographic correlators for \( \mathcal{N} \) = 4 SYM with general gauge group, JHEP 12 (2021) 159 [arXiv:2110.13106] [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
S. Collier and E. Perlmutter, Harnessing S-duality in \( \mathcal{N} \) = 4 SYM & supergravity as SL(2, ℤ)-averaged strings, JHEP 08 (2022) 195 [arXiv:2201.05093] [INSPIRE].
A. Brown, C. Wen and H. Xie, Generating functions and large-charge expansion of integrated correlators in \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory, arXiv:2303.17570 [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, tt* equations, localization and exact chiral rings in 4d \( \mathcal{N} \) = 2 SCFTs, JHEP 02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, Exact correlation functions in SU(2)\( \mathcal{N} \) = 2 superconformal QCD, Phys. Rev. Lett. 113 (2014) 251601 [arXiv:1409.4217] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, On exact correlation functions in SU(N) \( \mathcal{N} \) = 2 superconformal QCD, JHEP 11 (2015) 198 [arXiv:1508.03077] [INSPIRE].
M. D’Alessandro and L. Genovese, A Wide class of four point functions of BPS operators in N = 4 SYM at order g4, Nucl. Phys. B 732 (2006) 64 [hep-th/0504061] [INSPIRE].
D. Chicherin, J. Drummond, P. Heslop and E. Sokatchev, All three-loop four-point correlators of half-BPS operators in planar \( \mathcal{N} \) = 4 SYM, JHEP 08 (2016) 053 [arXiv:1512.02926] [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N = 4 SYM, Nucl. Phys. B 862 (2012) 450 [arXiv:1201.5329] [INSPIRE].
T. Fleury and R. Pereira, Non-planar data of \( \mathcal{N} \) = 4 SYM, JHEP 03 (2020) 003 [arXiv:1910.09428] [INSPIRE].
D. Chicherin, A. Georgoudis, V. Gonçalves and R. Pereira, All five-loop planar four-point functions of half-BPS operators in \( \mathcal{N} \) = 4 SYM, JHEP 11 (2018) 069 [arXiv:1809.00551] [INSPIRE].
J.L. Bourjaily, P. Heslop and V.-V. Tran, Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps, JHEP 11 (2016) 125 [arXiv:1609.00007] [INSPIRE].
L. Rastelli and X. Zhou, Mellin amplitudes for AdS5 × S5, Phys. Rev. Lett. 118 (2017) 091602 [arXiv:1608.06624] [INSPIRE].
L. Rastelli and X. Zhou, How to Succeed at Holographic Correlators Without Really Trying, JHEP 04 (2018) 014 [arXiv:1710.05923] [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. Caron-Huot and A.-K. Trinh, All tree-level correlators in AdS5 × S5 supergravity: hidden ten-dimensional conformal symmetry, JHEP 01 (2019) 196 [arXiv:1809.09173] [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].
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].
J.M. Drummond, H. Paul and M. Santagata, Bootstrapping string theory on AdS5 × S5, arXiv:2004.07282 [INSPIRE].
T. Abl, P. Heslop and A.E. Lipstein, Towards the Virasoro-Shapiro amplitude in AdS5 × S5, JHEP 04 (2021) 237 [arXiv:2012.12091] [INSPIRE].
F. Aprile, J.M. Drummond, H. Paul and M. Santagata, The Virasoro-Shapiro amplitude in AdS5 × S5 and level splitting of 10d conformal symmetry, JHEP 11 (2021) 109 [arXiv:2012.12092] [INSPIRE].
L.F. Alday, T. Hansen and J.A. Silva, AdS Virasoro-Shapiro from dispersive sum rules, JHEP 10 (2022) 036 [arXiv:2204.07542] [INSPIRE].
V. Gonçalves et al., Kaluza-Klein Five-Point Functions from AdS5 × S5 Supergravity, arXiv:2302.01896 [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].
L.F. Alday and S. Caron-Huot, Gravitational S-matrix from CFT dispersion relations, JHEP 12 (2018) 017 [arXiv:1711.02031] [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].
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].
J.M. Drummond, R. Glew and H. Paul, One-loop string corrections for AdS Kaluza-Klein amplitudes, JHEP 12 (2021) 072 [arXiv:2008.01109] [INSPIRE].
Z. Huang and E.Y. Yuan, Graviton Scattering in AdS5 × S5 at Two Loops, JHEP 04 (2023) 064 [arXiv:2112.15174] [INSPIRE].
J.M. Drummond and H. Paul, Two-loop supergravity on AdS5 × S5 from CFT, JHEP 08 (2022) 275 [arXiv:2204.01829] [INSPIRE].
C. Wen and S.-Q. Zhang, Integrated correlators in \( \mathcal{N} \) = 4 super Yang-Mills and periods, JHEP 05 (2022) 126 [arXiv:2203.01890] [INSPIRE].
Y. Hatsuda and K. Okuyama, Large N expansion of an integrated correlator in \( \mathcal{N} \) = 4 SYM, JHEP 11 (2022) 086 [arXiv:2208.01891] [INSPIRE].
Acknowledgments
The authors would like to thank Daniele Dorigoni, Michael Green and Rodolfo Russo for insightful discussions. CW is supported by Royal Society University Research Fellowships No. UF160350 and URF\R\221015. AB is supported by a Royal Society funding No. RF\ERE\210067.
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Brown, A., Wen, C. & Xie, H. Laplace-difference equation for integrated correlators of operators with general charges in \( \mathcal{N} \) = 4 SYM. J. High Energ. Phys. 2023, 66 (2023). https://doi.org/10.1007/JHEP06(2023)066
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DOI: https://doi.org/10.1007/JHEP06(2023)066