Abstract
It is known that the expectation value of Wilson loops in the Gross-Witten-Wadia (GWW) unitary matrix model can be computed exactly at finite N for arbitrary representations. We study the perturbative and non-perturbative corrections of Wilson loops in the 1/N expansion, either analytically or numerically using the exact result at finite N . As a by-product of the exact result of Wilson loops, we propose a large N master field of GWW model. This master field has an interesting eigenvalue distribution. We also study the Wilson loops in large representations, called Giant Wilson loops, and comment on the Hagedorn/deconfinement transition of a unitary matrix model with a double trace interaction.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Y. Hatsuda, S. Moriyama and K. Okuyama, Exact results on the ABJM Fermi gas, JHEP 10 (2012) 020 [arXiv:1207.4283] [INSPIRE].
Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton effects in ABJM theory from Fermi gas approach, JHEP 01 (2013) 158 [arXiv:1211.1251] [INSPIRE].
Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton bound states in ABJM theory, JHEP 05 (2013) 054 [arXiv:1301.5184] [INSPIRE].
Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, JHEP 09 (2014) 168 [arXiv:1306.1734] [INSPIRE].
D.J. Gross and E. Witten, Possible third order phase transition in the large N lattice gauge theory, Phys. Rev. D 21 (1980) 446 [INSPIRE].
S.R. Wadia, A study of U(N) lattice gauge theory in 2-dimensions, preprint EFI-79/44, University of Chicago, Chicago IL U.S.A., (1979) [arXiv:1212.2906] [INSPIRE].
V. Periwal and D. Shevitz, Unitary matrix models as exactly solvable string theories, Phys. Rev. Lett. 64 (1990) 1326 [INSPIRE].
I.R. Klebanov, J.M. Maldacena and N. Seiberg, Unitary and complex matrix models as 1D type 0 strings, Commun. Math. Phys. 252 (2004) 275 [hep-th/0309168] [INSPIRE].
M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [arXiv:0805.3033] [INSPIRE].
P.V. Buividovich, G.V. Dunne and S.N. Valgushev, Complex path integrals and saddles in two-dimensional gauge theory, Phys. Rev. Lett. 116 (2016) 132001 [arXiv:1512.09021] [INSPIRE].
G. Álvarez, L. Martínez Alonso and E. Medina, Complex saddles in the Gross-Witten-Wadia matrix model, Phys. Rev. D 94 (2016) 105010 [arXiv:1610.09948] [INSPIRE].
G. Grignani, J.L. Karczmarek and G.W. Semenoff, Hot giant loop holography, Phys. Rev. D 82 (2010) 027901 [arXiv:0904.3750] [INSPIRE].
J.L. Karczmarek, G.W. Semenoff and S. Yang, Comments on k-strings at large-N , JHEP 03 (2011) 075 [arXiv:1012.5875] [INSPIRE].
J.L. Karczmarek and G.W. Semenoff, Large representation recurrences in large-N random unitary matrix models, JHEP 10 (2011) 066 [arXiv:1108.2712] [INSPIRE].
B. Sundborg, The Hagedorn transition, deconfinement and N = 4 SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001] [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
H. Liu, Fine structure of Hagedorn transitions, hep-th/0408001 [INSPIRE].
I. Bars and F. Green, Complete integration of U(N ) lattice gauge theory in a large N limit, Phys. Rev. D 20 (1979) 3311 [INSPIRE].
P. Rossi, M. Campostrini and E. Vicari, The large-N expansion of unitary matrix models, Phys. Rept. 302 (1998) 143 [hep-lat/9609003] [INSPIRE].
Y.Y. Goldschmidt, 1/N expansion in two-dimensional lattice gauge theory, J. Math. Phys. 21 (1980) 1842 [INSPIRE].
F. Green and S. Samuel, Calculating the large-N phase transition in gauge and matrix models, Nucl. Phys. B 194 (1982) 107 [INSPIRE].
P. Rossi, On the exact evaluation of 〈det U p 〉 in a lattice gauge model, Phys. Lett. B 117 (1982) 72 [INSPIRE].
S. Mizoguchi, On unitary/Hermitian duality in matrix models, Nucl. Phys. B 716 (2005) 462 [hep-th/0411049] [INSPIRE].
N. Drukker and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP 02 (2005) 010 [hep-th/0501109] [INSPIRE].
S. Yamaguchi, Wilson loops of anti-symmetric representation and D5-branes, JHEP 05 (2006) 037 [hep-th/0603208] [INSPIRE].
S.A. Hartnoll and S.P. Kumar, Multiply wound Polyakov loops at strong coupling, Phys. Rev. D 74 (2006) 026001 [hep-th/0603190] [INSPIRE].
J. Gomis and F. Passerini, Holographic Wilson loops, JHEP 08 (2006) 074 [hep-th/0604007] [INSPIRE].
J. Gomis and F. Passerini, Wilson loops as D3-branes, JHEP 01 (2007) 097 [hep-th/0612022] [INSPIRE].
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn-deconfinement phase transition in weakly coupled large-N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
L. Álvarez-Gaumé, C. Gomez, H. Liu and S. Wadia, Finite temperature effective action, AdS 5 black holes and 1/N expansion, Phys. Rev. D 71 (2005) 124023 [hep-th/0502227] [INSPIRE].
S. Sachdev and J.-W. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, A simple model of quantum holography, talks at KITP, http://online.kitp.ucsb.edu/online/entangled15/kitaev, http://online.kitp.ucsb.edu/online/entangled15/kitaev2, University of California, Santa Barbara CA U.S.A., (2015).
R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [INSPIRE].
M. Hisakado, Unitary matrix models and Painlevé III, Mod. Phys. Lett. A 11 (1996) 3001 [hep-th/9609214] [INSPIRE].
C.A. Tracy and H. Widom, Random unitary matrices, permutations and Painlevé, Commun. Math. Phys. 207 (1999) 665 [math/9811154].
Digital Library of Mathematical Functions (DLMF) — eq. (10.19.8), http://dlmf.nist.gov/10.19.
Digital Library of Mathematical Functions (DLMF) — eq. (10.19.3), http://dlmf.nist.gov/10.19.
J. Ambjørn, L. Chekhov, C.F. Kristjansen and Yu. Makeenko, Matrix model calculations beyond the spherical limit, Nucl. Phys. B 404 (1993) 127 [Erratum ibid. B 449 (1995) 681] [hep-th/9302014] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1705.06542
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
Okuyama, K. Wilson loops in unitary matrix models at finite N . J. High Energ. Phys. 2017, 30 (2017). https://doi.org/10.1007/JHEP07(2017)030
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2017)030