Abstract
We compute the exact vacuum expectation value of circular Wilson loops for Euclidean \( \mathcal{N} \) = 4 super Yang-Mills with G = SO(N),Sp(N), in the fundamental and spinor representations. These field theories are dual to type IIB string theory compactified on AdS 5 × \( \mathbb{R}{\mathrm{\mathbb{P}}}^5 \) plus certain choices of discrete torsion, and we use our results to probe this holographic duality. We first revisit the LLM-type geometries having AdS 5 × \( \mathbb{R}{\mathrm{\mathbb{P}}}^5 \) as ground state. Our results clarify and refine the identification of these LLM-type geometries as bubbling geometries arising from fermions on a half harmonic oscillator. We furthermore identify the presence of discrete torsion with the one-fermion Wigner distribution becoming negative at the origin of phase space. We then turn to the string world-sheet interpretation of our results and argue that for the quantities considered they imply two features: first, the contribution coming from world-sheets with a single crosscap is closely related to the contribution coming from orientable world-sheets, and second, world-sheets with two crosscaps don’t contribute to these quantities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE].
N. Drukker and D.J. Gross, An exact prediction of N = 4 SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE].
J. Gomis, T. Okuda and D. Trancanelli, Quantum ’t Hooft operators and S-duality in N = 4 super Yang-Mills, Adv. Theor. Math. Phys. 13 (2009) 1941 [arXiv:0904.4486] [INSPIRE].
B. Fiol and G. Torrents, Exact results for Wilson loops in arbitrary representations, JHEP 01 (2014) 020 [arXiv:1311.2058] [INSPIRE].
S.-J. Rey and T. Suyama, Exact results and holography of Wilson loops in N = 2 superconformal (quiver) gauge theories, JHEP 01 (2011) 136 [arXiv:1001.0016] [INSPIRE].
F. Passerini and K. Zarembo, Wilson loops in N = 2 super-Yang-Mills from matrix model, JHEP 09 (2011) 102 [Erratum ibid. 10 (2011) 065] [arXiv:1106.5763] [INSPIRE].
B. Fraser and S.P. Kumar, Large rank Wilson loops in N = 2 superconformal QCD at strong coupling, JHEP 03 (2012) 077 [arXiv:1112.5182] [INSPIRE].
E. Witten, Baryons and branes in anti-de Sitter space, JHEP 07 (1998) 006 [hep-th/9805112] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
O. Aharony and E. Witten, Anti-de Sitter space and the center of the gauge group, JHEP 11 (1998) 018 [hep-th/9807205] [INSPIRE].
E. Witten, AdS/CFT correspondence and topological field theory, JHEP 12 (1998) 012 [hep-th/9812012] [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].
P. Caputa, R. de Mello Koch and P. Diaz, A basis for large operators in N = 4 SYM with orthogonal gauge group, JHEP 03 (2013) 041 [arXiv:1301.1560] [INSPIRE].
D. Correa, J. Henn, J. Maldacena and A. Sever, An exact formula for the radiation of a moving quark in N = 4 super Yang-Mills, JHEP 06 (2012) 048 [arXiv:1202.4455] [INSPIRE].
B. Fiol, B. Garolera and A. Lewkowycz, Exact results for static and radiative fields of a quark in N = 4 super Yang-Mills, JHEP 05 (2012) 093 [arXiv:1202.5292] [INSPIRE].
B. Fiol, B. Garolera and G. Torrents, Exact momentum fluctuations of an accelerated quark in N = 4 super Yang-Mills, JHEP 06 (2013) 011 [arXiv:1302.6991] [INSPIRE].
A. Lewkowycz and J. Maldacena, Exact results for the entanglement entropy and the energy radiated by a quark, JHEP 05 (2014) 025 [arXiv:1312.5682] [INSPIRE].
R.C. Myers and V. Periwal, The orientability of random surfaces, Phys. Rev. D 42 (1990) 3600 [INSPIRE].
A. Bilal, 2D gravity from matrix models: an introductory review, and particularities of antisymmetric matrix models, in Proceedings, Nonperturbative methods in low dimensional quantum field theories, Debrecen Hungary (1990), pg. 113 and CERN-TH-5867-90, CERN, Geneva Switzerland (1990) [INSPIRE].
N. Drukker and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP 02 (2005) 010 [hep-th/0501109] [INSPIRE].
S. Giombi, R. Ricci and D. Trancanelli, Operator product expansion of higher rank Wilson loops from D-branes and matrix models, JHEP 10 (2006) 045 [hep-th/0608077] [INSPIRE].
B. Fiol and B. Garolera, Energy loss of an infinitely massive half-Bogomol’nyi-Prasad-Sommerfeld particle by radiation to all orders in 1/N , Phys. Rev. Lett. 107 (2011) 151601 [arXiv:1106.5418] [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].
S. Mukhi and M. Smedback, Bubbling orientifolds, JHEP 08 (2005) 005 [hep-th/0506059] [INSPIRE].
G.M. Cicuta, Topological expansion for SO(N ) and Sp(2N ) gauge theories, Lett. Nuovo Cim. 35 (1982) 87 [INSPIRE].
S.G. Naculich, H.A. Riggs and H.J. Schnitzer, Two-dimensional Yang-Mills theories are string theories, Mod. Phys. Lett. A 8 (1993) 2223 [hep-th/9305097] [INSPIRE].
S. Ramgoolam, Comment on two-dimensional O(N ) and Sp(N ) Yang-Mills theories as string theories, Nucl. Phys. B 418 (1994) 30 [hep-th/9307085] [INSPIRE].
P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2D gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].
M. Mariño, Les Houches lectures on matrix models and topological strings, hep-th/0410165 [INSPIRE].
I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products. Academic Press, U.S.A. (2007).
T. Okuda and D. Trancanelli, Spectral curves, emergent geometry and bubbling solutions for Wilson loops, JHEP 09 (2008) 050 [arXiv:0806.4191] [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].
D. Berenstein, A toy model for the AdS/CFT correspondence, JHEP 07 (2004) 018 [hep-th/0403110] [INSPIRE].
G. Mandal, Fermions from half-BPS supergravity, JHEP 08 (2005) 052 [hep-th/0502104] [INSPIRE].
L. Grant, L. Maoz, J. Marsano, K. Papadodimas and V.S. Rychkov, Minisuperspace quantization of ‘bubbling AdS’ and free fermion droplets, JHEP 08 (2005) 025 [hep-th/0505079] [INSPIRE].
L. Maoz and V.S. Rychkov, Geometry quantization from supergravity: the case of ‘bubbling AdS’, JHEP 08 (2005) 096 [hep-th/0508059] [INSPIRE].
A. Royer, Wigner function as the expectation value of a parity operator, Phys. Rev. A 15 (1977) 449 [INSPIRE].
Y. Takayama and A. Tsuchiya, Complex matrix model and fermion phase space for bubbling AdS geometries, JHEP 10 (2005) 004 [hep-th/0507070] [INSPIRE].
V. Balasubramanian, J. de Boer, V. Jejjala and J. Simon, The library of Babel: on the origin of gravitational thermodynamics, JHEP 12 (2005) 006 [hep-th/0508023] [INSPIRE].
G. Nogues et al., Measurement of a negative value for the Wigner function of radiation, Phys. Rev. A 62 (2000) 054101.
R.L. Mkrtchian, The equivalence of Sp(2N ) and SO(−2N ) gauge theories, Phys. Lett. B 105 (1981) 174 [INSPIRE].
P. Cvitanovic and A.D. Kennedy, Spinors in negative dimensions, Phys. Scripta 26 (1982) 5 [INSPIRE].
R. Dijkgraaf and C. Vafa, A perturbative window into nonperturbative physics, hep-th/0208048 [INSPIRE].
H. Ita, H. Nieder and Y. Oz, Perturbative computation of glueball superpotentials for SO(N ) and USp(N ), JHEP 01 (2003) 018 [hep-th/0211261] [INSPIRE].
S.K. Ashok, R. Corrado, N. Halmagyi, K.D. Kennaway and C. Romelsberger, Unoriented strings, loop equations and N = 1 superpotentials from matrix models, Phys. Rev. D 67 (2003) 086004 [hep-th/0211291] [INSPIRE].
R.A. Janik and N.A. Obers, SO(N ) superpotential, Seiberg-Witten curves and loop equations, Phys. Lett. B 553 (2003) 309 [hep-th/0212069] [INSPIRE].
S. Sinha and C. Vafa, SO and Sp Chern-Simons at large-N , hep-th/0012136 [INSPIRE].
M. Mariño, String theory and the Kauffman polynomial, Commun. Math. Phys. 298 (2010) 613 [arXiv:0904.1088] [INSPIRE].
V. Bouchard, B. Florea and M. Mariño, Counting higher genus curves with crosscaps in Calabi-Yau orientifolds, JHEP 12 (2004) 035 [hep-th/0405083] [INSPIRE].
V. Bouchard, B. Florea and M. Mariño, Topological open string amplitudes on orientifolds, JHEP 02 (2005) 002 [hep-th/0411227] [INSPIRE].
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: 1406.5129
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Fiol, B., Garolera, B. & Torrents, G. Exact probes of orientifolds. J. High Energ. Phys. 2014, 169 (2014). https://doi.org/10.1007/JHEP09(2014)169
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2014)169