Abstract
We explain how the ’t Hooft expansion of correlators of half-BPS operators can be resummed in a large-charge limit in \( \mathcal{N} \) = 4 super Yang-Mills theory. The full correlator in the limit is given by a non-trivial function of two variables: One variable is the charge of the BPS operators divided by the square root of the number Nc of colors; the other variable is the octagon that contains all the ’t Hooft coupling and spacetime dependence. At each genus g in the large Nc expansion, this function is a polynomial of degree 2g + 2 in the octagon. We find several dual matrix model representations of the correlators in the large-charge limit. Amusingly, the number of colors in these matrix models is formally taken to zero in the relevant limit.
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References
F. Coronado, Perturbative four-point functions in planar \( \mathcal{N} \) = 4 SYM from hexagonalization, JHEP01 (2019) 056 [arXiv:1811.00467] [INSPIRE].
F. Coronado, Bootstrapping the simplest correlator in planar \( \mathcal{N} \) = 4 SYM at all loops, arXiv:1811.03282 [INSPIRE].
I. Kostov, V.B. Petkova and D. Serban, Determinant Formula for the Octagon Form Factor in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett.122 (2019) 231601 [arXiv:1903.05038] [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].
N.R. Constable et al., PP wave string interactions from perturbative Yang-Mills theory, JHEP07 (2002) 017 [hep-th/0205089] [INSPIRE].
N. Beisert, C. Kristjansen, J. Plefka, G.W. Semenoff and M. Staudacher, BMN correlators and operator mixing in N = 4 superYang-Mills theory, Nucl. Phys.B 650 (2003) 125 [hep-th/0208178] [INSPIRE].
N.R. Constable, D.Z. Freedman, M. Headrick and S. Minwalla, Operator mixing and the BMN correspondence, JHEP10 (2002) 068 [hep-th/0209002] [INSPIRE].
N. Beisert, C. Kristjansen, J. Plefka and M. Staudacher, BMN gauge theory as a quantum mechanical system, Phys. Lett.B 558 (2003) 229 [hep-th/0212269] [INSPIRE].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 superYang-Mills,JHEP04 (2002) 013 [hep-th/0202021] [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions, JHEP01 (2017) 130 [arXiv:1611.05577] [INSPIRE].
B. Eden and A. Sfondrini, Tessellating cushions: four-point functions in \( \mathcal{N} \) = 4 SYM, JHEP10 (2017) 098 [arXiv:1611.05436] [INSPIRE].
B. Eden, Y. Jiang, D. le Plat and A. Sfondrini, Colour-dressed hexagon tessellations for correlation functions and non-planar corrections, JHEP02 (2018) 170 [arXiv:1710.10212] [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions II: Two-Particle Contributions, JHEP02 (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].
T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling handles. Part II. Stratification and data analysis, JHEP11 (2018) 095 [arXiv:1809.09145] [INSPIRE].
H. Dorn, N. Drukker, G. Jorjadze and C. Kalousios, Space-like minimal surfaces in AdS × S, JHEP04 (2010) 004 [arXiv:0912.3829] [INSPIRE].
V.A. Kazakov, Solvable matrix models, 2000, hep-th/0003064 [INSPIRE].
A. Zvonkin, Matrix integrals and map enumeration: An accessible introduction, Math. Comput. Modelling26 (1997) 281.
T.W. Brown, Complex matrix model duality, Phys. Rev.D 83 (2011) 085002 [arXiv:1009.0674] [INSPIRE].
P.G. de Gennes, Exponents for the excluded volume problem as derived by the Wilson method, Phys. Lett.A 38 (1972) 339 [INSPIRE].
P.G. de Gennes and T.A. Witten, Scaling Concepts in Polymer Physics, Phys. Today33 (1980) 51.
J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics, Cambridge University Press (1996).
D. Sherrington and S. Kirkpatrick, Solvable Model of a Spin-Glass, Phys. Rev. Lett.35 (1975) 1792 [INSPIRE].
J.R.L. de Almeida and D.J. Thouless, Stability of the Sherrington-Kirkpatrick solution of a spin glass model, J. Phys.A 11 (1978) 983.
F. Morone, F. Caltagirone, E. Harrison and G. Parisi, Replica Theory and Spin Glasses, arXiv:1409.2722.
E. Brézin and S. Hikami, Characteristic Polynomials of Random Matrices, Commun. Math. Phys.214 (2000) 111.
E. Brézin and S. Hikami, Vertices from replica in a random matrix theory, J. Phys.A 40 (2007) 3545 [arXiv:0704.2044] [INSPIRE].
E. Brézin and S. Hikami, Intersection theory from duality and replica, Commun. Math. Phys.283 (2008) 507 [arXiv:0708.2210] [INSPIRE].
E. Brézin and S. Hikami, Duality and replicas for a unitary matrix model, JHEP07 (2010) 067 [arXiv:1005.4730] [INSPIRE].
S. Bellucci and C. Sochichiu, On matrix models for anomalous dimensions of super Yang-Mills theory, Nucl. Phys.B 726 (2005) 233 [hep-th/0410010] [INSPIRE].
P. Di Francesco, Rectangular matrix models and combinatorics of colored graphs, Nucl. Phys.B 648 (2003) 461 [cond-mat/0208037] [INSPIRE].
S.H. Shenker, The Strength of nonperturbative effects in string theory, in: The Large N expansion in quantum field theory and statistical physics: From spin systems to two-dimensional gravity, E. Brézin and S.R. Wadia, eds., World Scientific (1993), p. 809.
J. Polchinski, Combinatorics of boundaries in string theory, Phys. Rev.D 50 (1994) R6041 [hep-th/9407031] [INSPIRE].
J.M. Maldacena, G.W. Moore, N. Seiberg and D. Shih, Exact vs. semiclassical target space of the minimal string, JHEP10 (2004) 020 [hep-th/0408039] [INSPIRE].
R. Gopakumar, Open-Closed-Open String Duality, talk at the Johannesburg workshop: ‘Correlation Functions and the AdS/CFT Correspondence’, April 27, 2010, http://neo.phys.wits.ac.za/workshop 2/pdfs/rajesh.pdf.
P. Di Francesco and C. Itzykson, A generating function for fatgraphs, Ann. Inst. H. Poincaré Phys. Theor.59 (1993) 117 [hep-th/9212108] [INSPIRE].
A. Mironov and A. Morozov, On the complete perturbative solution of one-matrix models, Phys. Lett.B 771 (2017) 503 [arXiv:1705.00976] [INSPIRE].
I.K. Kostov, O(n) Vector Model on a Planar Random Lattice: Spectrum of Anomalous Dimensions, Mod. Phys. Lett.A 4 (1989) 217 [INSPIRE].
T. Bargheer, F. Coronado, V. Gonçalves and P. Vieira, Octagons II: Strong Coupling, to appear.
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP01 (2017) 013 [arXiv:1509.03612] [INSPIRE].
R.A. Janik, P. Surowka and A. Wereszczynski, On correlation functions of operators dual to classical spinning string states, JHEP05 (2010) 030 [arXiv:1002.4613] [INSPIRE].
T. Klose and T. McLoughlin, A light-cone approach to three-point functions in AdS 5 × S 5, JHEP04 (2012) 080 [arXiv:1106.0495] [INSPIRE].
J.A. Minahan, Holographic three-point functions for short operators, JHEP07 (2012) 187 [arXiv:1206.3129] [INSPIRE].
T. Bargheer, J.A. Minahan and R. Pereira, Computing Three-Point Functions for Short Operators, JHEP03 (2014) 096 [arXiv:1311.7461] [INSPIRE].
I.K. Kostov and M. Staudacher, Two-dimensional chiral matrix models and string theories, Phys. Lett.B 394 (1997) 75 [hep-th/9611011] [INSPIRE].
V.A. Kazakov, Exactly solvable potts models, bond and tree like percolation on dynamical (random) planar lattice, in: Field Theory on the Lattice. International Symposium, Seillac, France, September 28 - October 2, 1987, Nucl. Phys. Proc. Suppl.B 4 (1988) 93.
V.A. Kazakov, M. Staudacher and T. Wynter, Character expansion methods for matrix models of dually weighted graphs, Commun. Math. Phys.177 (1996) 451 [hep-th/9502132] [INSPIRE].
V.A. Kazakov, M. Staudacher and T. Wynter, Almost flat planar diagrams, Commun. Math. Phys.179 (1996) 235 [hep-th/9506174] [INSPIRE].
V.A. Kazakov, M. Staudacher and T. Wynter, Exact solution of discrete two-dimensional R 2gravity, Nucl. Phys.B 471 (1996) 309 [hep-th/9601069] [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, JHEP02 (2008) 030 [arXiv:0711.0176] [INSPIRE].
Ö. Gürdoğan and V. Kazakov, New Integrable 4D Quantum Field Theories from Strongly Deformed Planar \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett.117 (2016) 201602 [arXiv:1512.06704] [INSPIRE].
J. Caetano, Ö. Gürdoğan and V. Kazakov, Chiral limit of \( \mathcal{N} \) = 4 SYM and ABJM and integrable Feynman graphs, JHEP03 (2018) 077 [arXiv:1612.05895] [INSPIRE].
B. Basso and L.J. Dixon, Gluing Ladder Feynman Diagrams into Fishnets, Phys. Rev. Lett.119 (2017) 071601 [arXiv:1705.03545] [INSPIRE].
B. Basso, J. Caetano and T. Fleury, Hexagons and Correlators in the Fishnet Theory, arXiv:1812.09794 [INSPIRE].
N. Berkovits, Sketching a Proof of the Maldacena Conjecture at Small Radius, JHEP06 (2019) 111 [arXiv:1903.08264] [INSPIRE].
N. Gromov and A. Sever, The Holographic Fishchain, Phys. Rev. Lett.123 (2019) 081602 [arXiv:1903.10508] [INSPIRE].
V.A. Kazakov, Field theory as a matrix model, Nucl. Phys.B 587 (2000) 645 [hep-th/0003065] [INSPIRE].
A. Hatcher, On triangulations of surfaces, Topology Appl.40 (1991) 189.
J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math.85 (1986) 457.
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Bargheer, T., Coronado, F. & Vieira, P. Octagons I: combinatorics and non-planar resummations. J. High Energ. Phys. 2019, 162 (2019). https://doi.org/10.1007/JHEP08(2019)162
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DOI: https://doi.org/10.1007/JHEP08(2019)162