Abstract
We compute non-extremal three-point functions of scalar operators in \( \mathcal{N} \) = 4 super Yang-Mills at tree-level in gYM and at finite Nc, using the operator basis of the restricted Schur characters. We make use of the diagrammatic methods called quiver calculus to simplify the three-point functions. The results involve an invariant product of the generalized Racah-Wigner tensors (6j symbols). Assuming that the invariant product is written by the Littlewood-Richardson coefficients, we show that the non-extremal three- point functions satisfy the large Nc background independence; correspondence between the string excitations on AdS5 × S5 and those in the LLM geometry.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions, JHEP 01 (2017) 130 [arXiv:1611.05577] [INSPIRE].
B. Eden and A. Sfondrini, Tessellating cushions: four-point functions in \( \mathcal{N} \) = 4 SYM, JHEP 10 (2017) 098 [arXiv:1611.05436] [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions II: Two-Particle Contributions, JHEP 02 (2018) 177 [arXiv:1711.05327] [INSPIRE].
T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling Handles: Nonplanar Integrability in \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 121 (2018) 231602 [arXiv:1711.05326] [INSPIRE].
B. Eden, Y. Jiang, D. le Plat and A. Sfondrini, Colour-dressed hexagon tessellations for correlation functions and non-planar corrections, JHEP 02 (2018) 170 [arXiv:1710.10212] [INSPIRE].
T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling handles. Part II. Stratification and data analysis, JHEP 11 (2018) 095 [arXiv:1809.09145] [INSPIRE].
F. Coronado, Perturbative four-point functions in planar \( \mathcal{N} \) = 4 SYM from hexagonalization, JHEP 01 (2019) 056 [arXiv:1811.00467] [INSPIRE].
F. Coronado, Bootstrapping the simplest correlator in planar \( \mathcal{N} \) = 4 SYM at all loops, Phys. Rev. Lett. 124 (2020) 171601 [arXiv:1811.03282] [INSPIRE].
T. Bargheer, F. Coronado and P. Vieira, Octagons I: Combinatorics and Non-Planar Resummations, JHEP 08 (2019) 162 [arXiv:1904.00965] [INSPIRE].
T. Bargheer, F. Coronado and P. Vieira, Octagons II: Strong Coupling, arXiv:1909.04077 [INSPIRE].
B. Basso, V. Goncalves, S. Komatsu and P. Vieira, Gluing Hexagons at Three Loops, Nucl. Phys. B 907 (2016) 695 [arXiv:1510.01683] [INSPIRE].
B. Eden and A. Sfondrini, Three-point functions in \( \mathcal{N} \) = 4 SYM: the hexagon proposal at three loops, JHEP 02 (2016) 165 [arXiv:1510.01242] [INSPIRE].
B. Basso, V. Goncalves and S. Komatsu, Structure constants at wrapping order, JHEP 05 (2017) 124 [arXiv:1702.02154] [INSPIRE].
B. Eden, Y. Jiang, M. de Leeuw, T. Meier, D. le Plat and A. Sfondrini, Positivity of hexagon perturbation theory, JHEP 11 (2018) 097 [arXiv:1806.06051] [INSPIRE].
M. De Leeuw, B. Eden, D. Le Plat, T. Meier and A. Sfondrini, Multi-particle finite-volume effects for hexagon tessellations, arXiv:1912.12231 [INSPIRE].
R. Suzuki, Multi-trace Correlators from Permutations as Moduli Space, JHEP 05 (2019) 168 [arXiv:1810.09478] [INSPIRE].
S. Corley, A. Jevicki and S. Ramgoolam, Exact correlators of giant gravitons from dual N = 4 SYM theory, Adv. Theor. Math. Phys. 5 (2002) 809 [hep-th/0111222] [INSPIRE].
T.W. Brown, P.J. Heslop and S. Ramgoolam, Diagonal multi-matrix correlators and BPS operators in N = 4 SYM, JHEP 02 (2008) 030 [arXiv:0711.0176] [INSPIRE].
T.W. Brown, P.J. Heslop and S. Ramgoolam, Diagonal free field matrix correlators, global symmetries and giant gravitons, JHEP 04 (2009) 089 [arXiv:0806.1911] [INSPIRE].
Y. Kimura, S. Ramgoolam and R. Suzuki, Flavour singlets in gauge theory as Permutations, JHEP 12 (2016) 142 [arXiv:1608.03188] [INSPIRE].
R. de Mello Koch, J. Smolic and M. Smolic, Giant Gravitons — with Strings Attached (I), JHEP 06 (2007) 074 [hep-th/0701066] [INSPIRE].
R. de Mello Koch, J. Smolic and M. Smolic, Giant Gravitons — with Strings Attached (II), JHEP 09 (2007) 049 [hep-th/0701067] [INSPIRE].
D. Bekker, R. de Mello Koch and M. Stephanou, Giant Gravitons — with Strings Attached. III., JHEP 02 (2008) 029 [arXiv:0710.5372] [INSPIRE].
R. Bhattacharyya, S. Collins and R. de Mello Koch, Exact Multi-Matrix Correlators, JHEP 03 (2008) 044 [arXiv:0801.2061] [INSPIRE].
R. Bhattacharyya, R. de Mello Koch and M. Stephanou, Exact Multi-Restricted Schur Polynomial Correlators, JHEP 06 (2008) 101 [arXiv:0805.3025] [INSPIRE].
S. Collins, Restricted Schur Polynomials and Finite N Counting, Phys. Rev. D 79 (2009) 026002 [arXiv:0810.4217] [INSPIRE].
J. Pasukonis and S. Ramgoolam, Quivers as Calculators: Counting, Correlators and Riemann Surfaces, JHEP 04 (2013) 094 [arXiv:1301.1980] [INSPIRE].
P. Kramer, Orbital Fractional Parentage Coefficients for the Harmonic Oscillator Shell Model, Z. Phys. 205 (1967) 181.
R.W. Hasse and P.H. Butler, Symmetric and unitary group representations: I. Duality theory, J. Phys. A 17 (1984) 61.
N. Drukker and J. Plefka, Superprotected n-point correlation functions of local operators in N = 4 super Yang-Mills, JHEP 04 (2009) 052 [arXiv:0901.3653] [INSPIRE].
L.F. McAven, P.H. Butler and A.M. Hamel, Split bases and multiplicity separations in symmetric group transformation coefficients, J. Phys. A 31 (1998) 8363.
J.P. Elliott, J. Hope and H.A. Jahn, Theoretical Studies in Nuclear Structure IV. Wave Functions for the Nuclear p-shell Part B. Fractional Parentage Coefficients, Phil. Trans. Roy. Soc. A 246 (1953) 241.
J.-Q. Chen, D.F. Collinson and M.-J. Gao, Transformation coefficients of permutation groups, J. Math. Phys. 24 (198) 2695.
F. Pan and J.-Q. Chen, Irreducible representations of Hecke algebras in the non-standard basis and subduction coefficients, J. Phys. A 26 (1993) 4299.
L.F. McAven and P.H. Butler, Split-standard transformation coefficients: the block-selective conjecture, J. Phys. A 32 (1999) 7509.
L.F. McAven and A.M. Hamel, Calculating symmetric group split-standard transformation coefficients using the block selective method: a proof, J. Phys. A 35 (2002) 1719.
V. Chilla, On the linear equation method for the subduction problem in symmetric groups, J. Phys. A 39 (2006) 7657 [math-ph/0512011] [INSPIRE].
V. Chilla, Reduced subduction graph and higher multiplicity in Sn transformation coefficients, J. Phys. A 39 (2006) 12395 [math-ph/0606037] [INSPIRE].
R. de Mello Koch, N. Ives and M. Stephanou, On subgroup adapted bases for representations of the symmetric group, J. Phys. A 45 (2012) 135204 [arXiv:1112.4316] [INSPIRE].
R. de Mello Koch, M. Dessein, D. Giataganas and C. Mathwin, Giant Graviton Oscillators, JHEP 10 (2011) 009 [arXiv:1108.2761] [INSPIRE].
R. de Mello Koch, J.-H. Huang and L. Tribelhorn, Exciting LLM Geometries, JHEP 07 (2018) 146 [arXiv:1806.06586] [INSPIRE].
H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [INSPIRE].
M. Kim and H.J.R. van Zyl, Semiclassical SL(2) strings on LLM backgrounds, Phys. Lett. B 784 (2018) 62 [arXiv:1805.12460] [INSPIRE].
P. Mattioli and S. Ramgoolam, Permutation Centralizer Algebras and Multi-Matrix Invariants, Phys. Rev. D 93 (2016) 065040 [arXiv:1601.06086] [INSPIRE].
J. Ginibre, Statistical Ensembles of Complex, Quaternion and Real Matrices, J. Math. Phys. 6 (1965) 440 [INSPIRE].
M.L. Mehta, Random Matrices, third edition, Elsevier, Amsterdam The Netherlands (2004).
I.K. Kostov and M. Staudacher, Two-dimensional chiral matrix models and string theories, Phys. Lett. B 394 (1997) 75 [hep-th/9611011] [INSPIRE].
I.K. Kostov, M. Staudacher and T. Wynter, Complex matrix models and statistics of branched coverings of 2 − D surfaces, Commun. Math. Phys. 191 (1998) 283 [hep-th/9703189] [INSPIRE].
C. Kristjansen, J. Plefka, G.W. Semenoff and M. Staudacher, A New double scaling limit of N = 4 superYang-Mills theory and PP wave strings, Nucl. Phys. B 643 (2002) 3 [hep-th/0205033] [INSPIRE].
A.N. Kirillov and N.Y. Reshetikhin, Representations of the algebra Uq (sl(2)), q-orthogonal polynomials and invariants of links, in New Developments in the Theory of Knots, World Scientific, Singapore (1990), pg. 202.
H. Itoyama, A. Mironov, A. Morozov and A. Morozov, Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations, Int. J. Mod. Phys. A 28 (2013) 1340009 [arXiv:1209.6304] [INSPIRE].
S. Nawata, P. Ramadevi and Zodinmawia, Multiplicity-free quantum 6-j symbols for Uq (slN ), Lett. Math. Phys. 103 (2013) 1389 [arXiv:1302.5143] [INSPIRE].
A. Morozov and A. Sleptsov, New symmetries for the Uq (slN ) 6-j symbols from the Eigenvalue conjecture, JETP Lett. 108 (2018) 697 [arXiv:1905.01876] [INSPIRE].
J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from Anti-de Sitter space, JHEP 06 (2000) 008 [hep-th/0003075] [INSPIRE].
V. Balasubramanian, M. Berkooz, A. Naqvi and M.J. Strassler, Giant gravitons in conformal field theory, JHEP 04 (2002) 034 [hep-th/0107119] [INSPIRE].
R. de Mello Koch, C. Mathwin and H.J.R. van Zyl, LLM Magnons, JHEP 03 (2016) 110 [arXiv:1601.06914] [INSPIRE].
R. de Mello Koch, M. Kim and H.J.R. Zyl, Integrable Subsectors from Holography, JHEP 05 (2018) 198 [arXiv:1802.01367] [INSPIRE].
H. Takayanagi and T. Takayanagi, Notes on giant gravitons on PP waves, JHEP 12 (2002) 018 [hep-th/0209160] [INSPIRE].
S. Hirano and Y. Sato, Giant graviton interactions and M2-branes ending on multiple M5-branes, JHEP 05 (2018) 065 [arXiv:1803.04172] [INSPIRE].
R. de Mello Koch, E. Gandote and J.-H. Huang, Non-Perturbative String Theory from AdS/CFT, JHEP 02 (2019) 169 [arXiv:1901.02591] [INSPIRE].
A. Bissi, C. Kristjansen, D. Young and K. Zoubos, Holographic three-point functions of giant gravitons, JHEP 06 (2011) 085 [arXiv:1103.4079] [INSPIRE].
P. Caputa, R. de Mello Koch and K. Zoubos, Extremal versus Non-Extremal Correlators with Giant Gravitons, JHEP 08 (2012) 143 [arXiv:1204.4172] [INSPIRE].
H. Lin, Giant gravitons and correlators, JHEP 12 (2012) 011 [arXiv:1209.6624] [INSPIRE].
C. Kristjansen, S. Mori and D. Young, On the Regularization of Extremal Three-point Functions Involving Giant Gravitons, Phys. Lett. B 750 (2015) 379 [arXiv:1507.03965] [INSPIRE].
Y. Jiang, S. Komatsu and E. Vescovi, Exact Three-Point Functions of Determinant Operators in Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 123 (2019) 191601 [arXiv:1907.11242] [INSPIRE].
Y. Jiang, S. Komatsu and E. Vescovi, Structure Constants in \( \mathcal{N} \) = 4 SYM at Finite Coupling as Worldsheet g-Function, arXiv:1906.07733 [INSPIRE].
G. Chen, R. de Mello Koch, M. Kim and H.J.R. Van Zyl, Absorption of closed strings by giant gravitons, JHEP 10 (2019) 133 [arXiv:1908.03553] [INSPIRE].
K.-Y. Kim, M. Kim and K. Lee, Structure Constants of a Single Trace Operator and Determinant Operators from Hexagon, arXiv:1906.11515 [INSPIRE].
D.M. Goldschmidt, University Lecture Series. Vol. 4: Group Characters, Symmetric Functions and the Hecke Algebras, AMS Press, Providence U.S.A. (1993).
R.P. Stanley, Cambridge Studies in Advanced Mathematics. Book 62: Enumerative Combinatorics: Volume 2, Cambridge University Press, Cambridge U.K. (1999).
E.P. Wigner, Group theory and its application to the quantum mechanics of atomic spectra, Academic Press, New York U.S.A. (1959).
G. Racah, Theory of Complex Spectra. II, Phys. Rev. 62 (1942) 438 [INSPIRE].
P. Kramer, Recoupling Coefficients of the Symmetric Group for Shell and Cluster Model Configurations, Z. Phys. 216 (1968) 68.
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
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: 2002.07216
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Suzuki, R. Three-point functions in \( \mathcal{N} \) = 4 SYM at finite Nc and background independence. J. High Energ. Phys. 2020, 118 (2020). https://doi.org/10.1007/JHEP05(2020)118
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2020)118