Abstract
In favourable cases the low energy dynamics of a stack of M2-branes at a toric Calabi-Yau fourfold singularity can be described by an \( \mathcal{N} = 2 \) supersymmetric Chern- Simons quiver theory, but there still does not exists an “inverse algorithm” going from the toric data of the CY 4 to the CS quiver. We make progress in that direction by deriving CS quiver theories for M2-branes probing cones over a large class of geometries Y p, q (B 4), which are \( {S^{{3}}}/{\mathbb{Z}_{\text{p}}} \) bundles over toric Fano varieties B 4. We rely on the type IIA understanding of CS quivers, giving a firm string theory footing to our CS theories. In particular we give a derivation of some previously conjectured CS quivers in the case \( {B_{{4}}} = \mathbb{C}{\mathbb{P}^{{1}}} \times \mathbb{C}{\mathbb{P}^{{1}}} \), as field theories dual to M-theory backgrounds with nontrivial torsion G 4 fluxes.
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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] [INSPIRE].
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) 1133 ] [hep-th/9711200] [INSPIRE].
B.S. Acharya, J. Figueroa-O’Farrill, C. Hull and B.J. Spence, Branes at conical singularities and holography, Adv. Theor. Math. Phys. 2 (1999) 1249 [hep-th/9808014] [INSPIRE].
D.L. Jafferis and A. Tomasiello, A Simple class of N = 3 gauge/gravity duals, JHEP 10 (2008) 101 [arXiv:0808.0864] [INSPIRE].
D. Martelli and J. Sparks, Moduli spaces of Chern-Simons quiver gauge theories and AdS 4 /CF T 3, Phys. Rev. D 78 (2008) 126005 [arXiv:0808.0912] [INSPIRE].
A. Hanany and A. Zaffaroni, Tilings, Chern-Simons Theories and M2 Branes, JHEP 10 (2008) 111 [arXiv:0808.1244] [INSPIRE].
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] [INSPIRE].
K. Ueda and M. Yamazaki, Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories, JHEP 12 (2008) 045 [arXiv:0808.3768] [INSPIRE].
Y. Imamura and K. Kimura, Quiver Chern-Simons theories and crystals, JHEP 10 (2008) 114 [arXiv:0808.4155] [INSPIRE].
A. Hanany, D. Vegh and A. Zaffaroni, Brane Tilings and M2 Branes, JHEP 03 (2009) 012 [arXiv:0809.1440] [INSPIRE].
M. Aganagic, A Stringy Origin of M2 Brane Chern-Simons Theories, Nucl. Phys. B 835 (2010) 1 [arXiv:0905.3415] [INSPIRE].
D.L. Jafferis, Quantum corrections to N = 2 Chern-Simons theories with flavor and their AdS 4 duals, arXiv:0911.4324 [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Chiral flavors and M2-branes at toric CY4 singularities, JHEP 02 (2010) 036 [arXiv:0911.4127] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS 4 /CF T 3, JHEP 09 (2011) 005 [arXiv:1105.2299] [INSPIRE].
D. Gang, C. Hwang, S. Kim and J. Park, Tests of AdS 4 /CFT 3 correspondence for \( \mathcal{N} = 2 \) chiral-like theory, JHEP 02 (2012) 079 [arXiv:1111.4529] [INSPIRE].
D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, arXiv:1012.3210 [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
S. Cheon, H. Kim and N. Kim, Calculating the partition function of N = 2 Gauge theories on S 3 and AdS/CFT correspondence, JHEP 05 (2011) 134 [arXiv:1102.5565] [INSPIRE].
D. Martelli and J. Sparks, The large-N limit of quiver matrix models and Sasaki-Einstein manifolds, Phys. Rev. D 84 (2011) 046008 [arXiv:1102.5289] [INSPIRE].
S. Cheon, D. Gang, S. Kim and J. Park, Refined test of AdS 4 /CF T 3 correspondence for N =2,3 theories, JHEP 05(2011) 027 [arXiv:1102.4273][INSPIRE].
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, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
A. Amariti, C. Klare and M. Siani, The Large-N Limit of Toric Chern-Simons Matter Theories and Their Duals, arXiv:1111.1723 [INSPIRE].
A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [INSPIRE].
S. Franco, A. Hanany, K.D. Kennaway, D. Vegh and B. Wecht, Brane dimers and quiver gauge theories, JHEP 01 (2006) 096 [hep-th/0504110] [INSPIRE].
J.P. Gauntlett, D. Martelli, J.F. Sparks and D. Waldram, A New infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys. 8 (2006) 987 [hep-th/0403038] [INSPIRE].
D. Martelli and J. Sparks, Notes on toric Sasaki-Einstein seven-manifolds and AdS 4 /CFT 3, JHEP 11 (2008) 016 [arXiv:0808.0904] [INSPIRE].
A. Futaki, H. Ono and G. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Diff. Geom. 83 (2009) 585 [math/0607586] [INSPIRE]
A. Hanany and R.-K. Seong, Brane Tilings and Reflexive Polygons, arXiv:1201.2614 [INSPIRE].
A.D. King, Moduli of representations of finite-dimensional algebras, Q. J. Math. 45 (1994) 515 .
A. Tomasiello and A. Zaffaroni, Parameter spaces of massive IIA solutions, JHEP 04 (2011) 067 [arXiv:1010.4648] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].
C. Closset, Seiberg duality for Chern-Simons quivers and D-brane mutations, JHEP 03 (2012) 056 [arXiv:1201.2432] [INSPIRE].
I. Wolfram Research, Mathematica Edition: Version 7.0, Wolfram Research, Inc. (2008).
J. Davey, A. Hanany and J. Pasukonis, On the Classification of Brane Tilings, JHEP 01 (2010) 078 [arXiv:0909.2868] [INSPIRE].
M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
D.-E. Diaconescu, M.R. Douglas and J. Gomis, Fractional branes and wrapped branes, JHEP 02 (1998) 013 [hep-th/9712230] [INSPIRE].
A.E. Lawrence, N. Nekrasov and C. Vafa, On conformal field theories in four-dimensions, Nucl. Phys. B 533 (1998) 199 [hep-th/9803015] [INSPIRE].
I.R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [INSPIRE].
D.R. Morrison and M.R. Plesser, Nonspherical horizons. 1., Adv. Theor. Math. Phys. 3 (1999)1 [hep-th/9810201] [INSPIRE].
C. Beasley, B.R. Greene, C. Lazaroiu and M. Plesser, D3-branes on partial resolutions of Abelian quotient singularities of Calabi-Yau threefolds, Nucl. Phys. B 566 (2000) 599 [hep-th/9907186] [INSPIRE].
B. Feng, A. Hanany and Y.-H. He, D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 [hep-th/0003085] [INSPIRE].
B. Feng, S. Franco, A. Hanany and Y.-H. He, UnHiggsing the del Pezzo, JHEP 08 (2003) 058 [hep-th/0209228] [INSPIRE].
K.D. Kennaway, Brane Tilings, Int. J. Mod. Phys. A 22 (2007) 2977 [arXiv:0706.1660] [INSPIRE].
D. Marolf, Chern-Simons terms and the three notions of charge, hep-th/0006117 [INSPIRE].
A. Hanany and D. Vegh, Quivers, tilings, branes and rhombi, JHEP 10 (2007) 029 [hep-th/0511063] [INSPIRE].
D.R. Gulotta, Properly ordered dimers, R-charges and an efficient inverse algorithm, JHEP 10 (2008) 014 [arXiv:0807.3012] [INSPIRE].
S. Franco and D. Vegh, Moduli spaces of gauge theories from dimer models: Proof of the correspondence, JHEP 11 (2006) 054 [hep-th/0601063] [INSPIRE].
M.A. Luty and W. Taylor, Varieties of vacua in classical supersymmetric gauge theories, Phys. Rev. D 53 (1996) 3399 [hep-th/9506098] [INSPIRE].
D. Martelli and J. Sparks, Symmetry-breaking vacua and baryon condensates in AdS/CFT, Phys. Rev. D 79 (2009) 065009 [arXiv:0804.3999] [INSPIRE].
S. Mozgovoy, Crepant resolutions and brane tilings I: Toric realization, arXiv:0908.3475.
M. Bender and S. Mozgovoy, Crepant resolutions and brane tilings II: Tilting bundles, arXiv:0909.2013.
A. Hanany, C.P. Herzog and D. Vegh, Brane tilings and exceptional collections, JHEP 07 (2006) 001 [hep-th/0602041] [INSPIRE].
P.S. Aspinwall, D-Branes on Toric Calabi-Yau Varieties, arXiv:0806.2612 [INSPIRE].
D. Cox, J. Little, and H. Schenck, Toric Varieties, American Mathematical Society (2011).
W. Stein et. al., Sage Mathematics Software (Version 4.7.2), The Sage Development Team (2011), http://www.sagemath.org.
R. Minasian and G.W. Moore, K theory and Ramond-Ramond charge, JHEP 11 (1997) 002 [hep-th/9710230] [INSPIRE].
P.S. Aspinwall, D-branes on Calabi-Yau manifolds, hep-th/0403166 [INSPIRE].
C.P. Herzog, Seiberg duality is an exceptional mutation, JHEP 08 (2004) 064 [hep-th/0405118] [INSPIRE].
D.S. Freed and E. Witten, Anomalies in string theory with D-branes, Asian J. Math 3 (1999) 819 [hep-th/9907189] [INSPIRE].
M.R. Douglas, B. Fiol and C. Romelsberger, Stability and BPS branes, JHEP 09 (2005) 006 [hep-th/0002037] [INSPIRE].
P.S. Aspinwall, D-branes, Pi-stability and theta-stability, hep-th/0407123 [INSPIRE].
P.S. Aspinwall and I.V. Melnikov, D-branes on vanishing del Pezzo surfaces, JHEP 12 (2004) 042 [hep-th/0405134] [INSPIRE].
O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].
O. Aharony, A. Hashimoto, S. Hirano and P. Ouyang, D-brane Charges in Gravitational Duals of 2 + 1 Dimensional Gauge Theories and Duality Cascades, JHEP 01 (2010) 072 [arXiv:0906.2390] [INSPIRE].
D. Gaiotto and D.L. Jafferis, Notes on adding D6 branes wrapping Rp 3 in AdS 4 × CP 3, arXiv:0903.2175 [INSPIRE].
J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS backgrounds in string and M-theory, hep-th/0411194 [INSPIRE].
W. Chen, H. Lü, C. Pope and J.F. Vazquez-Poritz, A Note on Einstein Sasaki metrics in D ≥ 7, Class. Quant. Grav. 22 (2005) 3421 [hep-th/0411218] [INSPIRE].
O. Aharony, IR duality in D = 3 N = 2 supersymmetric USp(2 N(c)) and U(N(c)) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].
S. Cremonesi, Type IIB construction of flavoured ABJ(M) and fractional M2 branes, JHEP 01 (2011) 076 [arXiv:1007.4562] [INSPIRE].
S. Cremonesi, Higgsing chiral M2-brane theories and decompactification of D6-branes, work in progress.
M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
M. Aganagic and K. Schaeffer, Wall Crossing, Quivers and Crystals, arXiv:1006.2113 [INSPIRE].
J. Davey, A. Hanany, N. Mekareeya and G. Torri, Phases of M2-brane Theories, JHEP 06 (2009) 025 [arXiv:0903.3234] [INSPIRE].
A. Butti, D. Forcella and A. Zaffaroni, Counting BPS baryonic operators in CFTs with Sasaki-Einstein duals, JHEP 06 (2007) 069 [hep-th/0611229] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
F. Benini, F. Canoura, S. Cremonesi, C. Núñez and A.V. Ramallo, Backreacting flavors in the Klebanov-Strassler background, JHEP 09 (2007) 109 [arXiv:0706.1238] [INSPIRE].
O. Bergman and S. Hirano, Anomalous radius shift in AdS 4 /CF T 3, JHEP 07 (2009) 016 [arXiv:0902.1743] [INSPIRE].
O. Aharony, D. Jafferis, A. Tomasiello and A. Zaffaroni, Massive type IIA string theory cannot be strongly coupled, JHEP 11 (2010) 047 [arXiv:1007.2451] [INSPIRE].
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Closset, C., Cremonesi, S. Toric Fano varieties and Chern-Simons quivers. J. High Energ. Phys. 2012, 60 (2012). https://doi.org/10.1007/JHEP05(2012)060
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DOI: https://doi.org/10.1007/JHEP05(2012)060