Abstract
We recently proved that, when integrating out the spacetime dependence with a certain integration measure, four-point correlators \( \left\langle {\mathcal{O}}_2{\mathcal{O}}_2{\mathcal{O}}_p^{(i)}{\mathcal{O}}_p^{(i)}\right\rangle \) in 𝒩 = 4 supersymmetric Yang-Mills theory with SU(N) gauge group are governed by a universal Laplace-difference equation. Here \( {\mathcal{O}}_p^{(i)} \) is a superconformal primary with charge p and degeneracy i. These physical observables, called integrated correlators, are modular-invariant functions of Yang-Mills coupling τ. The Laplace-difference equation is a recursion relation that relates integrated correlators of operators with different charges. In this paper, we introduce the generating functions for these integrated correlators that sum over the charge. By utilising the Laplace-difference equation, we determine the generating functions for all the integrated correlators, in terms of the initial data of the recursion relation. We show that the transseries of the integrated correlators in the large-p (i.e. large-charge) expansion for a fixed N consists of three parts: 1) is independent of τ, which behaves as a power series in 1/p, plus an additional log(p) term when i = j; 2) is a power series in 1/p, with coefficients given by a sum of the non-holomorphic Eisenstein series; 3) is a sum of exponentially decayed modular functions in the large-p limit, which can be viewed as a generalisation of the non-holomorphic Eisenstein series. When i = j, there is an additional modular function of τ that is independent of p and is fully determined in terms of the integrated correlator with p = 2. The Laplace-difference equation was obtained with a reorganisation of the operators that means the large-charge limit is taken in a particular way here. From these SL(2, ℤ)-invariant results, we also determine the generalised ’t Hooft genus expansion and the associated large-p non-perturbative corrections of the integrated correlators by introducing λ = p \( {g}_{YM}^2 \). The generating functions have subtle differences between even and odd N, which have important consequences in the large-charge expansion and resurgence analysis. We also consider the generating functions of the integrated correlators for some fixed p by summing over N, and we study their large-N behaviour, as well as comment on the similarities and differences between the large-p expansion and the large-N expansion.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Dorigoni, M.B. Green and C. Wen, Novel representation of an integrated correlator in 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 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 N = 4 SYM with general classical gauge groups, SciPost Phys. 13 (2022) 092 [arXiv:2202.05784] [INSPIRE].
D. Dorigoni, M.B. Green, C. Wen and H. Xie, Modular-invariant large-N completion of an integrated correlator in N = 4 supersymmetric Yang-Mills theory, JHEP 04 (2023) 114 [arXiv:2210.14038] [INSPIRE].
D.J. Binder, S.M. Chester, S.S. Pufu and Y. Wang, N = 4 super-Yang-Mills correlators at strong coupling from string theory and localization, JHEP 12 (2019) 119 [arXiv:1902.06263] [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].
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 N = 4 super-Yang-Mills, JHEP 11 (2020) 016 [arXiv:1912.13365] [INSPIRE].
S.M. Chester and S.S. Pufu, Far beyond the planar limit in strongly-coupled N = 4 SYM, JHEP 01 (2021) 103 [arXiv:2003.08412] [INSPIRE].
S.M. Chester et al., New modular invariants in N = 4 super-Yang-Mills theory, JHEP 04 (2021) 212 [arXiv:2008.02713] [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].
L.F. Alday, S.M. Chester and T. Hansen, Modular invariant holographic correlators for N = 4 SYM with general gauge group, JHEP 12 (2021) 159 [arXiv:2110.13106] [INSPIRE].
C. Montonen and D.I. Olive, Magnetic monopoles as gauge particles?, Phys. Lett. B 72 (1977) 117 [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge theories and magnetic charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
K.A. Intriligator, Bonus symmetries of N = 4 super-Yang-Mills correlation functions via AdS duality, Nucl. Phys. B 551 (1999) 575 [hep-th/9811047] [INSPIRE].
K.A. Intriligator and W. Skiba, Bonus symmetry and the operator product expansion of N = 4 super-Yang-Mills, Nucl. Phys. B 559 (1999) 165 [hep-th/9905020] [INSPIRE].
H. Paul, E. Perlmutter and H. Raj, Integrated correlators in N = 4 SYM via SL(2, Z) spectral theory, JHEP 01 (2023) 149 [arXiv:2209.06639] [INSPIRE].
S. Collier and E. Perlmutter, Harnessing S-duality in 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, Laplace-difference equation for integrated correlators of operators with general charges in N = 4 SYM, JHEP 06 (2023) 066 [arXiv:2303.13195] [INSPIRE].
E. D’Hoker et al., Extremal correlators in the AdS/CFT correspondence, hep-th/9908160 [https://doi.org/10.1142/9789812793850_0020] [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] [INSPIRE].
C. Rayson, Some aspects of conformal N = 4 SYM four point function, Ph.D. thesis, Cambridge U., Cambridge, U.K. (2008) [arXiv:1706.04450] [INSPIRE].
S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT operator spectrum at large global charge, JHEP 12 (2015) 071 [arXiv:1505.01537] [INSPIRE].
A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, Goldstone bosons and CFT data, JHEP 06 (2017) 011 [arXiv:1611.02912] [INSPIRE].
L. Alvarez-Gaume, O. Loukas, D. Orlando and S. Reffert, Compensating strong coupling with large charge, JHEP 04 (2017) 059 [arXiv:1610.04495] [INSPIRE].
S. Hellerman, S. Maeda and M. Watanabe, Operator dimensions from moduli, JHEP 10 (2017) 089 [arXiv:1706.05743] [INSPIRE].
D. Jafferis, B. Mukhametzhanov and A. Zhiboedov, Conformal bootstrap at large charge, JHEP 05 (2018) 043 [arXiv:1710.11161] [INSPIRE].
S. Hellerman et al., Universal correlation functions in rank 1 SCFTs, JHEP 12 (2019) 047 [arXiv:1804.01535] [INSPIRE].
L.Á. Gaumé, D. Orlando and S. Reffert, Selected topics in the large quantum number expansion, Phys. Rept. 933 (2021) 1 [arXiv:2008.03308] [INSPIRE].
S. Hellerman and S. Maeda, On the large R-charge expansion in N = 2 superconformal field theories, JHEP 12 (2017) 135 [arXiv:1710.07336] [INSPIRE].
A. Grassi, Z. Komargodski and L. Tizzano, Extremal correlators and random matrix theory, JHEP 04 (2021) 214 [arXiv:1908.10306] [INSPIRE].
A. Bourget, D. Rodriguez-Gomez and J.G. Russo, A limit for large R-charge correlators in N = 2 theories, JHEP 05 (2018) 074 [arXiv:1803.00580] [INSPIRE].
M. Beccaria, On the large R-charge N = 2 chiral correlators and the Toda equation, JHEP 02 (2019) 009 [arXiv:1809.06280] [INSPIRE].
M. Beccaria, Double scaling limit of N = 2 chiral correlators with Maldacena-Wilson loop, JHEP 02 (2019) 095 [arXiv:1810.10483] [INSPIRE].
S. Hellerman et al., S-duality and correlation functions at large R-charge, JHEP 04 (2021) 287 [arXiv:2005.03021] [INSPIRE].
S. Hellerman and D. Orlando, Large R-charge EFT correlators in N = 2 SQCD, arXiv:2103.05642 [INSPIRE].
S. Hellerman, On the exponentially small corrections to N = 2 superconformal correlators at large R-charge, arXiv:2103.09312 [INSPIRE].
G. Cuomo and Z. Komargodski, Giant vortices and the Regge limit, JHEP 01 (2023) 006 [arXiv:2210.15694] [INSPIRE].
H. Paul, E. Perlmutter and H. Raj, Exact large charge in N = 4 SYM and semiclassical string theory, arXiv:2303.13207 [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].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
E. Gerchkovitz et al., Correlation functions of Coulomb branch operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].
F. Aprile et al., Single particle operators and their correlators in free N = 4 SYM, JHEP 11 (2020) 072 [arXiv:2007.09395] [INSPIRE].
Y. Hatsuda and K. Okuyama, Large N expansion of an integrated correlator in N = 4 SYM, JHEP 11 (2022) 086 [arXiv:2208.01891] [INSPIRE].
C. Luo and Y. Wang, Casimir energy and modularity in higher-dimensional conformal field theories, JHEP 07 (2023) 028 [arXiv:2212.14866] [INSPIRE].
G.V. Dunne and M. Unsal, Deconstructing zero: resurgence, supersymmetry and complex saddles, JHEP 12 (2016) 002 [arXiv:1609.05770] [INSPIRE].
C. Kozçaz, T. Sulejmanpasic, Y. Tanizaki and M. Ünsal, Cheshire cat resurgence, self-resurgence and quasi-exact solvable systems, Commun. Math. Phys. 364 (2018) 835 [arXiv:1609.06198] [INSPIRE].
D. Dorigoni and P. Glass, The grin of Cheshire cat resurgence from supersymmetric localization, SciPost Phys. 4 (2018) 012 [arXiv:1711.04802] [INSPIRE].
D. Dorigoni and P. Glass, Picard-Lefschetz decomposition and Cheshire cat resurgence in 3D N = 2 field theories, JHEP 12 (2019) 085 [arXiv:1909.05262] [INSPIRE].
T. Fujimori and P. Glass, Resurgence in 2-dimensional Yang-Mills and a genus-altering deformation, PTEP 2023 (2023) 053B03 [arXiv:2212.11988] [INSPIRE].
D. Dorigoni and A. Kleinschmidt, Modular graph functions and asymptotic expansions of Poincaré series, Commun. Num. Theor. Phys. 13 (2019) 569 [arXiv:1903.09250] [INSPIRE].
D. Dorigoni and A. Kleinschmidt, Resurgent expansion of Lambert series and iterated Eisenstein integrals, Commun. Num. Theor. Phys. 15 (2021) 1 [arXiv:2001.11035] [INSPIRE].
D. Dorigoni, A. Kleinschmidt and R. Treilis, To the cusp and back: resurgent analysis for modular graph functions, JHEP 11 (2022) 048 [arXiv:2208.14087] [INSPIRE].
F. Aprile and P. Vieira, Large p explorations. From SUGRA to big STRINGS in Mellin space, JHEP 12 (2020) 206 [arXiv:2007.09176] [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.
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: 2303.17570
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
Brown, A., Wen, C. & Xie, H. Generating functions and large-charge expansion of integrated correlators in 𝒩 = 4 supersymmetric Yang-Mills theory. J. High Energ. Phys. 2023, 129 (2023). https://doi.org/10.1007/JHEP07(2023)129
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2023)129