Abstract
We test the AdS/CFT correspondence by computing the partition function of some \( \mathcal{N} = 2 \) quiver Chern-Simons-matter theories on three-sphere. The M-theory backgrounds are of the Freund-Rubin type with the seven-dimensional internal space given as Sasaki-Einstein manifolds Q 1,1,1 or V 5,2. Localization technique reduces the exact path integral to a matrix model, and we study the large-N behavior of the partition function. For simplicity we consider only non-chiral models which have a real-valued partition function. The result is in full agreement with the prediction of the gravity duals, i.e. the free energy is proportional to N 3/2 and the coefficient matches correctly the volume of Q 1,1,1 and V 5,2.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [SPIRES].
D. Martelli and J. Sparks, Notes on toric Sasaki-Einstein seven-manifolds and AdS 4 /CFT 3, JHEP 11 (2008) 016 [arXiv:0808.0904] [SPIRES].
D. Martelli and J. Sparks, Moduli spaces of Chern-Simons quiver gauge theories and AdS 4 /CFT 3, Phys. Rev. D 78 (2008) 126005 [arXiv:0808.0912] [SPIRES].
A. Hanany and A. Zaffaroni, Tilings, Chern-Simons theories and M2 branes, JHEP 10 (2008) 111 [arXiv:0808.1244] [SPIRES].
K. Ueda and M. Yamazaki, Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories, JHEP 12 (2008) 045 [arXiv:0808.3768] [SPIRES].
Y. Imamura and K. Kimura, Quiver Chern-Simons theories and crystals, JHEP 10 (2008) 114 [arXiv:0808.4155] [SPIRES].
A. Hanany, D. Vegh and A. Zaffaroni, Brane tilings and M2 branes, JHEP 03 (2009) 012 [arXiv:0809.1440] [SPIRES].
S. Franco, A. Hanany, J. Park and D. Rodriguez-Gomez, Towards M2-brane theories for generic toric singularities, JHEP 12 (2008) 110 [arXiv:0809.3237] [SPIRES].
M. Aganagic, A stringy origin of M2 brane Chern-Simons theories, Nucl. Phys. B 835 (2010) 1 [arXiv:0905.3415] [SPIRES].
D. Martelli and J. Sparks, AdS 4 /CFT 3 duals from M2-branes at hypersurface singularities and their deformations, JHEP 12 (2009) 017 [arXiv:0909.2036] [SPIRES].
D.L. Jafferis, Quantum corrections to N = 2 Chern-Simons theories with flavor and their AdS 4 duals, arXiv:0911.4324 [SPIRES].
F. Benini, C. Closset and S. Cremonesi, Chiral flavors and M2-branes at toric CY4 singularities, JHEP 02 (2010) 036 [arXiv:0911.4127] [SPIRES].
D. Martelli and J. Sparks, Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals, Commun. Math. Phys. 262 (2006) 51 [hep-th/0411238] [SPIRES].
S. Franco et al., Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [SPIRES].
S. Lee, Superconformal field theories from crystal lattices, Phys. Rev. D 75 (2007) 101901 [hep-th/0610204] [SPIRES].
S. Lee, S. Lee and J. Park, Toric AdS 4 /CFT 3 duals and M-theory crystals, JHEP 05 (2007) 004 [hep-th/0702120] [SPIRES].
S. Kim, S. Lee, S. Lee and J. Park, Abelian gauge theory on M2-brane and toric duality, Nucl. Phys. B 797 (2008) 340 [arXiv:0705.3540] [SPIRES].
J. Bhattacharya and S. Minwalla, Superconformal indices for \( \mathcal{N} = 6 \) Chern-Simons theories, JHEP 01 (2009) 014 [arXiv:0806.3251] [SPIRES].
S. Kim, The complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [arXiv:0903.4172] [SPIRES].
Y. Imamura and S. Yokoyama, A monopole index for N = 4 Chern-Simons theories, Nucl. Phys. B 827 (2010) 183 [arXiv:0908.0988] [SPIRES].
Y. Imamura and S. Yokoyama, T wisted sectors in gravity duals of N = 4 Chern-Simons theories, JHEP 11 (2010) 059 [arXiv:1008.3180] [SPIRES].
S. Kim and J. Park, Probing AdS 4 /CFT 3 proposals beyond chiral rings, JHEP 08 (2010) 069 [arXiv:1003.4343] [SPIRES].
Y. Imamura, D. Yokoyama and S. Yokoyama, Superconformal index for large-N quiver Chern-Simons theories, arXiv:1102.0621 [SPIRES].
S. Cheon, D. Gang, S. Kim and J. Park, Refined test of AdS 4 /CFT 3 correspondence for N = 2, 3 theories, JHEP 05 (2011) 027 [arXiv:1102.4273] [SPIRES].
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] [SPIRES].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, arXiv:1007.3837 [SPIRES].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [SPIRES].
M. Mariño and P. Putrov, Exact results in ABJM theory from topological strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [SPIRES].
N. Halmagyi and V. Yasnov, The spectral curve of the lens space matrix model, JHEP 11 (2009) 104 [hep-th/0311117] [SPIRES].
T. Suyama, On large-N solution of Gaiotto-Tomasiello theory, JHEP 10 (2010) 101 [arXiv:1008.3950] [SPIRES].
C.P. Herzog, I.R. Klebanov, S.S. Pufu and T. Tesileanu, Multi-matrix models and tri-Sasaki Einstein spaces, Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [SPIRES].
D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, arXiv:1012.3210 [SPIRES].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [SPIRES].
S. Kharchev, D. Lebedev and M. Semenov-Tian-Shansky, Unitary representations of \( {U_q}\left( {\mathfrak{s}\mathfrak{l}\left( {2,\mathbb{R}} \right)} \right) \) , the modular double and the multiparticle q-deformed Toda chains, Commun. Math. Phys. 225 (2002) 573 [hep-th/0102180] [SPIRES].
A.G. Bytsko and J. Teschner, Quantization of models with non-compact quantum group symmetry: modular XXZ magnet and lattice sinh-Gordon model, J. Phys. A 39 (2006) 12927 [hep-th/0602093] [SPIRES].
D. Martelli and J. Sparks, The large-N limit of quiver matrix models and Sasaki-Einstein manifolds, arXiv:1102.5289 [SPIRES].
D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem: N = 2 field theories on the three-sphere, arXiv:1103.1181 [SPIRES].
J.H. Lee, S. Kim, J. Kim and N. Kim, Probing non-toric geometry with rotating membranes, Nucl. Phys. B 838 (2010) 238 [arXiv:1003.6111] [SPIRES].
S. Franco, I.R. Klebanov and D. Rodriguez-Gomez, M2-branes on orbifolds of the cone over Q 1,1,1, JHEP 08 (2009) 033 [arXiv:0903.3231] [SPIRES].
D. Fabbri et al., 3D superconformal theories from Sasakian seven-manifolds: new nontrivial evidences for AdS 4 /CFT 3, Nucl. Phys. B 577 (2000) 547 [hep-th/9907219] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1102.5565
Rights and permissions
About this article
Cite this article
Cheon, S., Kim, H. & Kim, N. Calculating the partition function of \( \mathcal{N} = 2 \) gauge theories on S 3 and AdS/CFT correspondence. J. High Energ. Phys. 2011, 134 (2011). https://doi.org/10.1007/JHEP05(2011)134
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2011)134