Abstract
We study the n-point functions of scalar multi-trace operators in the U(Nc) gauge theory with adjacent scalars, such as \( \mathcal{N} \) = 4 super Yang-Mills, at tree-level by using finite group methods. We derive a set of formulae of the general n-point functions, valid for general n and to all orders of 1/Nc. In one formula, the sum over Feynman graphs becomes a topological partition function on Σ0,n with a discrete gauge group, which resembles closed string interactions. In another formula, a new skeleton reduction of Feynman graphs generates connected ribbon graphs, which resembles open string interaction. We define the moduli space \( {\mathrm{\mathcal{M}}}_{g,n}^{\mathrm{gauge}} \) from the space of skeleton-reduced graphs in the connected n-point function of gauge theory. This moduli space is a proper subset of \( {\mathrm{\mathcal{M}}}_{g,n} \) stratified by the genus, and its top component gives a simple triangulation of Σg,n.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].
N. Beisert, C. Kristjansen and M. Staudacher, The dilatation operator of conformal N = 4 super Yang-Mills theory, Nucl. Phys. B 664 (2003) 131 [hep-th/0303060] [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure constants and integrable bootstrap in planar N = 4 SYM theory, arXiv:1505.06745 [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: nonplanar integrability in 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, JHEP 11 (2018) 095 [arXiv:1809.09145] [INSPIRE].
N. Beisert, C. Kristjansen, J. Plefka, G.W. Semenoff and M. Staudacher, BMN correlators and operator mixing in N = 4 super Yang-Mills theory, Nucl. Phys. B 650 (2003) 125 [hep-th/0208178] [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].
R. Bhattacharyya, S. Collins and R. de Mello Koch, Exact multi-matrix correlators, JHEP 03 (2008) 044 [arXiv:0801.2061] [INSPIRE].
J. Pasukonis and S. Ramgoolam, Quivers as calculators: counting, correlators and Riemann surfaces, JHEP 04 (2013) 094 [arXiv:1301.1980] [INSPIRE].
E. Brézin, C. Itzykson, G. Parisi and J.B. Zuber, Planar diagrams, Commun. Math. Phys. 59 (1978) 35 [INSPIRE].
E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys Diff. Geom. 1 (1991) 243 [INSPIRE].
M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys. 147 (1992) 1 [INSPIRE].
J. Jenkins, On the existence of certain general extremal metrics, Ann. Math. 66 (1957) 440.
K. Strebel, Quardatic differentials, Springer-Verlag, Berlin Heidelberg, Germany (1984).
J.L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157.
J.L. Harer, The cohomology of the moduli space of curves, Lect. Notes Math. 1337 (1988) 138.
M. Mulase and M. Penkava, Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over \( \overline{\mathbb{Q}} \), Asian J. Math. 2 (1998) 875 [math-ph/9811024].
M. Mulase and M. Penkava, Topological recursion for the Poincaré polynomial of the combinatorial moduli space of curves, arXiv:1009.2135 [INSPIRE].
P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. I.H.É.S. 36 (1969) 75.
E. D’Hoker and D.H. Phong, The geometry of string perturbation theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].
L. Chekhov, Matrix models and geometry of moduli spaces, hep-th/9509001 [INSPIRE].
R. Gopakumar, From free fields to AdS: III, Phys. Rev. D 72 (2005) 066008 [hep-th/0504229] [INSPIRE].
K. Furuuchi, From free fields to AdS: thermal case, Phys. Rev. D 72 (2005) 066009 [hep-th/0505148] [INSPIRE].
O. Aharony, Z. Komargodski and S.S. Razamat, On the worldsheet theories of strings dual to free large N gauge theories, JHEP 05 (2006) 016 [hep-th/0602226] [INSPIRE].
J.R. David and R. Gopakumar, From spacetime to worldsheet: four point correlators, JHEP 01 (2007) 063 [hep-th/0606078] [INSPIRE].
O. Aharony, J.R. David, R. Gopakumar, Z. Komargodski and S.S. Razamat, Comments on worldsheet theories dual to free large N gauge theories, Phys. Rev. D 75 (2007) 106006 [hep-th/0703141] [INSPIRE].
O. Aharony and Z. Komargodski, The space-time operator product expansion in string theory duals of field theories, JHEP 01 (2008) 064 [arXiv:0711.1174] [INSPIRE].
S.S. Razamat, On a worldsheet dual of the Gaussian matrix model, JHEP 07 (2008) 026 [arXiv:0803.2681] [INSPIRE].
S.S. Razamat, From matrices to strings and back, JHEP 03 (2010) 049 [arXiv:0911.0658] [INSPIRE].
S. Charbonnier, B. Eynard and F. David, Large Strebel graphs and (3, 2) Liouville CFT, Annales Henri Poincaré 19 (2018) 1611 [arXiv:1709.02709] [INSPIRE].
R. de Mello Koch and S. Ramgoolam, From matrix models and quantum fields to Hurwitz space and the absolute Galois group, arXiv:1002.1634 [INSPIRE].
R. Gopakumar, What is the simplest gauge-string duality?, arXiv:1104.2386 [INSPIRE].
R. Gopakumar and R. Pius, Correlators in the simplest gauge-string duality, JHEP 03 (2013) 175 [arXiv:1212.1236] [INSPIRE].
R. de Mello Koch and L. Nkumane, Topological string correlators from matrix models, JHEP 03 (2015) 004 [arXiv:1411.5226] [INSPIRE].
T. Eguchi and S.-K. Yang, The topological CP 1 model and the large N matrix integral, Mod. Phys. Lett. A 9 (1994) 2893 [hep-th/9407134] [INSPIRE].
C. Kristjansen, J. Plefka, G.W. Semenoff and M. Staudacher, A new double scaling limit of N = 4 super Yang-Mills theory and PP wave strings, Nucl. Phys. B 643 (2002) 3 [hep-th/0205033] [INSPIRE].
R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].
M. Fukuma, S. Hosono and H. Kawai, Lattice topological field theory in two-dimensions, Commun. Math. Phys. 161 (1994) 157 [hep-th/9212154] [INSPIRE].
S.-W. Chung, M. Fukuma and A.D. Shapere, Structure of topological lattice field theories in three-dimensions, Int. J. Mod. Phys. A 9 (1994) 1305 [hep-th/9305080] [INSPIRE].
G.W. Moore and G. Segal, D-branes and k-theory in 2D topological field theory, hep-th/0609042 [INSPIRE].
Y. Kimura, S. Ramgoolam and R. Suzuki, Flavour singlets in gauge theory as permutations, JHEP 12 (2016) 142 [arXiv:1608.03188] [INSPIRE].
M. Mulase and M. Penkava, Combinatorial structure of the moduli space of Riemann surfaces and the KP equations, https://www.math.ucdavis.edu/~mulase/texfiles/1997moduli.pdf, (1997).
G. Mondello, Riemann surfaces, ribbon graphs and combinatorial classes, in Handbook of Teichmüller theory, volume 2, (2007) [arXiv:0705.1792].
B. Eynard, Counting surfaces, Progr. Math. Phys. 70, Springer, Basel, Switzerland (2016).
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
S.B. Giddings and S.A. Wolpert, A triangulation of moduli space from light cone string theory, Commun. Math. Phys. 109 (1987) 177 [INSPIRE].
S. Nakamura, A calculation of the orbifold Euler number of the moduli space of curves by a new cell decomposition of the Teichmüller space, Tokyo J. Math. 23 (2000) 87.
L. Freidel, D. Garner and S. Ramgoolam, Permutation combinatorics of worldsheet moduli space, Phys. Rev. D 91 (2015) 126001 [arXiv:1412.3979] [INSPIRE].
D. Garner and S. Ramgoolam, The geometry of the light-cone cell decomposition of moduli space, J. Math. Phys. 56 (2015) 112301 [arXiv:1507.02968] [INSPIRE].
L. Hollands and A. Neitzke, Spectral networks and Fenchel-Nielsen coordinates, Lett. Math. Phys. 106 (2016) 811 [arXiv:1312.2979] [INSPIRE].
Y. Kazama and S. Komatsu, On holographic three point functions for GKP strings from integrability, JHEP 01 (2012) 110 [Erratum ibid. 06 (2012) 150] [arXiv:1110.3949] [INSPIRE].
Y. Kazama and S. Komatsu, Wave functions and correlation functions for GKP strings from integrability, JHEP 09 (2012) 022 [arXiv:1205.6060] [INSPIRE].
Y. Kazama and S. Komatsu, Three-point functions in the SU(2) sector at strong coupling, JHEP 03 (2014) 052 [arXiv:1312.3727] [INSPIRE].
Y. Kazama, S. Komatsu and T. Nishimura, Classical integrability for three-point functions: cognate structure at weak and strong couplings, JHEP 10 (2016) 042 [Erratum ibid. 02 (2018) 047] [arXiv:1603.03164] [INSPIRE].
L.F. Alday and J. Maldacena, Minimal surfaces in AdS and the eight-gluon scattering amplitude at strong coupling, arXiv:0903.4707 [INSPIRE].
L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in anti-de-Sitter space, JHEP 11 (2009) 082 [arXiv:0904.0663] [INSPIRE].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic bubble ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].
L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for scattering amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [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].
D. Rodriguez-Gómez and J.G. Russo, Large N correlation functions in superconformal field theories, JHEP 06 (2016) 109 [arXiv:1604.07416] [INSPIRE].
D. Rodriguez-Gomez and J.G. Russo, Operator mixing in large N superconformal field theories on S 4 and correlators with Wilson loops, JHEP 12 (2016) 120 [arXiv:1607.07878] [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].
R. de Mello Koch, J.-H. Huang and L. Tribelhorn, Exciting LLM geometries, JHEP 07 (2018) 146 [arXiv:1806.06586] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized S matrices in two-dimensions as the exact solutions of certain relativistic quantum field models, Annals Phys. 120 (1979) 253 [INSPIRE].
L.D. Faddeev, Quantum completely integral models of field theory, Sov. Sci. Rev. C 1 (1980) 107 [INSPIRE].
M. Kim, N. Kiryu, S. Komatsu and T. Nishimura, Structure constants of defect changing operators on the 1/2 BPS Wilson loop, JHEP 12 (2017) 055 [arXiv:1710.07325] [INSPIRE].
A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Quantum spectral curve and structure constants in N = 4 SYM: cusps in the ladder limit, JHEP 10 (2018) 060 [arXiv:1802.04237] [INSPIRE].
B. Frab and D. Margalit, A primer on mapping class groups, Princeton University Press, Princeton, NJ, U.S.A. (2012).
B. Eynard, Lectures notes on compact Riemann surfaces, arXiv:1805.06405.
B. Zwiebach, How covariant closed string theory solves a minimal area problem, Commun. Math. Phys. 136 (1991) 83 [INSPIRE].
R.C. Penner, Perturbative series and the moduli space of Riemann surfaces, J. Diff. Geom. 27 (1988) 35 [INSPIRE].
V.V. Fock, Description of moduli space of projective structures via fat graphs, hep-th/9312193 [INSPIRE].
S.K. Ashok, F. Cachazo and E. Dell’Aquila, Strebel differentials with integral lengths and Argyres-Douglas singularities, hep-th/0610080 [INSPIRE].
P. Caputa, C. Kristjansen and K. Zoubos, On the spectral problem of N = 4 SYM with orthogonal or symplectic gauge group, JHEP 10 (2010) 082 [arXiv:1005.2611] [INSPIRE].
P. Caputa, R. de Mello Koch and P. Diaz, Operators, correlators and free fermions for SO(N) and Sp(N), JHEP 06 (2013) 018 [arXiv:1303.7252] [INSPIRE].
G. Kemp, Restricted Schurs and correlators for SO(N) and Sp(N), JHEP 08 (2014) 137 [arXiv:1406.3854] [INSPIRE].
C. Lewis-Brown and S. Ramgoolam, BPS operators in N = 4 SO(N) super Yang-Mills theory: plethysms, dominoes and words, JHEP 11 (2018) 035 [arXiv:1804.11090] [INSPIRE].
J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].
P. Vieira, Gamma matrices and Wick contractions, in Mathematica summer school on theoretical physics, (2013).
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
ArXiv ePrint: 1810.09478
Electronic supplementary material
Below is the link to the electronic supplementary material.
ESM 1
(GZ 1513 kb)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Suzuki, R. Multi-trace correlators from permutations as moduli space. J. High Energ. Phys. 2019, 168 (2019). https://doi.org/10.1007/JHEP05(2019)168
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2019)168