Abstract
We propose that, in \( \mathcal{N} = 4 \) Super Yang-Mills theory, correlation functions of certain 1/8 BPS Wilson loops and local operators inserted on a S 2 in space-time may be computed in terms of analogous observables in the “zero-instanton” sector of 2d Yang-Mills theory. The Wilson loops are mapped to the standard Wilson loops of the 2d theory, as recently conjectured, while the local operators are mapped to powers of the 2d field strength. We give several perturbative checks of the correspondence, and derive from 2d Yang-Mills a two-matrix model for the correlator of a local operator and a Wilson loop of arbitrary shape. We show that the strong coupling planar limit of the two-matrix model precisely agrees with a string theory calculation in AdS 5×S 5
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References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [SPIRES].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [SPIRES].
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] [SPIRES].
N. Drukker and D.J. Gross, An exact prediction of N =4 SUSY M theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [SPIRES].
G.W. Semenoff and K. Zarembo, Wilson loops in SYM theory: from weak to strong coupling, Nucl. Phys. Proc. Suppl. 108 (2002) 106 [hep-th/0202156] [SPIRES].
N. Drukker and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP 02 (2005) 010 [hep-th/0501109] [SPIRES].
J. Gomis and F. Passerini, Holographic Wilson loops, JHEP 08 (2006) 074 [hep-th/0604007] [SPIRES].
S. Yamaguchi, Wilson loops of anti-symmetric representation and D5-branes, JHEP 05 (2006) 037 [hep-th/0603208] [SPIRES].
J. Gomis and F. Passerini, Wilson loops as D3-branes, JHEP 01 (2007) 097 [hep-th/0612022] [SPIRES].
S. Yamaguchi, Bubbling geometries for half BPS Wilson lines, Int. J. Mod. Phys. A 22 (2007) 1353 [hep-th/0601089] [SPIRES].
O. Lunin, On gravitational description of Wilson lines, JHEP 06 (2006) 026 [hep-th/0604133] [SPIRES].
E. D’Hoker, J. Estes and M. Gutperle, Gravity duals of half-BPS Wilson loops, JHEP 06 (2007) 063 [arXiv:0705.1004] [SPIRES].
T. Okuda, A prediction for bubbling geometries, JHEP 01 (2008) 003 [arXiv:0708.3393] [SPIRES].
T. Okuda and D. Trancanelli, Spectral curves, emergent geometry and bubbling solutions for Wilson loops, JHEP 09 (2008) 050 [arXiv:0806.4191] [SPIRES].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [SPIRES].
N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Supersymmetric Wilson loops on S 3, JHEP 05 (2008) 017 [arXiv:0711.3226] [SPIRES].
N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Wilson loops: From four-dimensional SYM to two-dimensional YM, Phys. Rev. D 77 (2008) 047901 [arXiv:0707.2699] [SPIRES].
N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, More supersymmetric Wilson loops, Phys. Rev. D 76 (2007) 107703 [arXiv:0704.2237] [SPIRES].
A.A. Migdal, Gauge transitions in gauge and spin lattice systems, Sov. Phys. JETP 42 (1975) 743 [SPIRES].
M. Blau and G. Thompson, Quantum Yang-Mills theory on arbitrary surfaces, Int. J. Mod. Phys. A7 (1992) 3781 [SPIRES].
M. Blau and G. Thompson, Lectures on 2-D gauge theories: topological aspects and path integral techniques, hep-th/9310144 [SPIRES].
E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991) 153 [SPIRES].
A. Bassetto and L. Griguolo, Two-dimensional QCD, instanton contributions and the perturbative Wu-Mandelstam-Leibbrandt prescription, Phys. Lett. B 443 (1998) 325 [hep-th/9806037] [SPIRES].
A. Bassetto, S. Nicoli and F. Vian, Topological contributions in two-dimensional Yang-Mills theory: From group averages to integration over algebras, Lett. Math. Phys. 57 (2001) 97 [hep-th/0101052] [SPIRES].
M. Staudacher and W. Krauth, Two-dimensional QCD in the Wu-Mandelstam-Leibbrandt prescription, Phys. Rev. D 57 (1998) 2456 [hep-th/9709101] [SPIRES].
A. Bassetto, L. Griguolo, F. Pucci and D. Seminara, Supersymmetric Wilson loops at two loops, JHEP 06 (2008) 083 [arXiv:0804.3973] [SPIRES].
D. Young, BPS Wilson loops on S 2 at higher loops, JHEP 05 (2008) 077 [arXiv:0804.4098] [SPIRES].
S. Giombi, V. Pestun and R. Ricci, Notes on supersymmetric Wilson loops on a two-sphere, JHEP 07 (2010) 088 [arXiv:0905.0665] [SPIRES].
A. Bassetto et al., Correlators of supersymmetric Wilson-loops, protected operators and matrix models in N =4 SYM, JHEP 08 (2009) 061 [arXiv:0905.1943] [SPIRES].
V. Pestun, Localization of the four-dimensional N =4 SYM to a two-sphere and 1/8 BPS Wilson loops, arXiv:0906.0638 [SPIRES].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [SPIRES].
A.A. Gerasimov and S.L. Shatashvili, Higgs bundles, gauge theories and quantum groups, Commun. Math. Phys. 277 (2008) 323 [hep-th/0609024] [SPIRES].
A.A. Gerasimov and S.L. Shatashvili, Two-dimensional gauge theories and quantum integrable systems, arXiv:0711.1472 [SPIRES].
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] [SPIRES].
G.W. Semenoff and K. Zarembo, More exact predictions of SUSYM for string theory, Nucl. Phys. B 616 (2001) 34 [hep-th/0106015] [SPIRES].
G.W. Semenoff and D. Young, Exact 1/4 BPS loop: Chiral primary correlator, Phys. Lett. B 643 (2006) 195 [hep-th/0609158] [SPIRES].
V. Pestun and K. Zarembo, Comparing strings in AdS 5×S 5 to planar diagrams: an example, Phys. Rev. D 67 (2003) 086007 [hep-th/0212296] [SPIRES].
K. Okuyama and G.W. Semenoff, Wilson loops in N =4 SYM and fermion droplets, JHEP 06 (2006) 057 [hep-th/0604209] [SPIRES].
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] [SPIRES].
K. Zarembo, Open string fluctuations in AdS 5×S 5 and operators with large R charge, Phys. Rev. D 66 (2002) 105021 [hep-th/0209095] [SPIRES].
J. Gomis, S. Matsuura, T. Okuda and D. Trancanelli, Wilson loop correlators at strong coupling: from matrices to bubbling geometries, JHEP 08 (2008) 068 [arXiv:0807.3330] [SPIRES].
N. Drukker, 1/4 BPS circular loops, unstable world-sheet instantons and the matrix model, JHEP 09 (2006) 004 [hep-th/0605151] [SPIRES].
P. de Medeiros, C.M. Hull, B.J. Spence and J.M. Figueroa-O’Farrill, Conformal topological Yang-Mills theory and de Sitter holography, JHEP 08 (2002) 055 [hep-th/0111190] [SPIRES].
J.M. Daul, V.A. Kazakov and I.K. Kostov, Rational theories of 2-D gravity from the two matrix model, Nucl. Phys. B 409 (1993) 311 [hep-th/9303093] [SPIRES].
D.E. Berenstein, R. Corrado, W. Fischler and J.M. Maldacena, The operator product expansion for Wilson loops and surfaces in the large-N limit, Phys. Rev. D 59 (1999) 105023 [hep-th/9809188] [SPIRES].
J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from Anti-de Sitter space, JHEP 06 (2000) 008 [hep-th/0003075] [SPIRES].
A. Hashimoto, S. Hirano and N. Itzhaki, Large branes in AdS and their field theory dual, JHEP 08 (2000) 051 [hep-th/0008016] [SPIRES].
H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [SPIRES].
S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three-point functions of chiral operators in D =4, N =4 SY M at large-N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [SPIRES].
H.J. Kim, L.J. Romans and P. van Nieuwenhuizen, The mass spectrum of chiral N =2 D = 10 supergravity on S 5 , Phys. Rev. D 32 (1985) 389 [SPIRES].
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ArXiv ePrint: 0906.1572
On leave of absence from ITEP, 117218, Moscow, Russia.
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Giombi, S., Pestun, V. Correlators of local operators and 1/8 BPS Wilson loops on S 2 from 2d YM and matrix models. J. High Energ. Phys. 2010, 33 (2010). https://doi.org/10.1007/JHEP10(2010)033
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DOI: https://doi.org/10.1007/JHEP10(2010)033