Abstract
Recently it was discovered that twisted superconformal index \( \mathcal{I} \) can be used to understand the Bekenstein-Hawking entropy of magnetically charged black holes in AdS spacetime. In this paper we apply the so-called \( \mathcal{I} \) -extremization procedure to three-dimensional gauge field theories and their geometric dual, focusing in particular on the seven-dimensional Sasaki-Einstein manifold M 1,1,1. We generalize recent studies on relations among toric geometry, variational principles, and black hole entropy to the case of AdS2 × Y9, where Y9 is a fibration of toric Sasaki-Einstein manifold M 1,1,1 over a two-dimensional Riemann surface Σg. The nine-dimensional variational problem is given in terms of an entropy functional. In order to illustrate the computations as explicitly as possible, we consider cases where either only mesonic or baryonic fluxes are turned on. By employing the operator counting method, we calculate the S3 free energy and the topologically twisted index \( \mathcal{I} \) at large-N. The result for \( \mathcal{I} \), it turns out, can be also obtained from the variational principle of the entropy functional with mesonic fluxes. We also study asymptotically AdS4 black holes which are magnetically charged with respect to the vector field in the Betti multiplet. By extremizing the entropy functional with baryonic flux, we compute the entropy and find that it agrees with the entropy of an explicit solution in a four-dimensional gauged supergravity which is a consistent truncation of eleven-dimensional supergravity in AdS4 × M 1,1,1.
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References
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.L. Cacciatori and D. Klemm, Supersymmetric AdS4 black holes and attractors, JHEP 01 (2010) 085 [arXiv:0911.4926] [INSPIRE].
G. Dall’Agata and A. Gnecchi, Flow equations and attractors for black holes in N = 2 U(1) gauged supergravity, JHEP 03 (2011) 037 [arXiv:1012.3756] [INSPIRE].
K. Hristov and S. Vandoren, Static supersymmetric black holes in AdS4 with spherical symmetry, JHEP 04 (2011) 047 [arXiv:1012.4314] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M 2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Exact microstate counting for dyonic black holes in AdS4, Phys. Lett. B 771 (2017) 462 [arXiv:1608.07294] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS5 black holes, JHEP 10 (2019) 062 [arXiv:1810.11442] [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv:1810.12067 [INSPIRE].
F. Benini and P. Milan, Black holes in 4d \( \mathcal{N} \) = 4 Super-Yang-Mills, arXiv:1812.09613 [INSPIRE].
A. Zaffaroni, Lectures on AdS black holes, holography and localization, arXiv:1902.07176 [INSPIRE].
C. Couzens, J.P. Gauntlett, D. Martelli and J. Sparks, A geometric dual of c-extremization, JHEP 01 (2019) 212 [arXiv:1810.11026] [INSPIRE].
J.P. Gauntlett, D. Martelli and J. Sparks, Toric geometry and the dual of c-extremization, JHEP 01 (2019) 204 [arXiv:1812.05597] [INSPIRE].
F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [INSPIRE].
F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].
S.M. Hosseini and A. Zaffaroni, Proving the equivalence of c-extremization and its gravitational dual for all toric quivers, JHEP 03 (2019) 108 [arXiv:1901.05977] [INSPIRE].
F. Azzurli, N. Bobev, P.M. Crichigno, V.S. Min and A. Zaffaroni, A universal counting of black hole microstates in AdS4, JHEP 02 (2018) 054 [arXiv:1707.04257] [INSPIRE].
I.R. Klebanov, S.S. Pufu and T. Tesileanu, Membranes with topological charge and AdS4 /CFT3 correspondence, Phys. Rev. D 81 (2010) 125011 [arXiv:1004.0413] [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].
S.M. Hosseini and A. Zaffaroni, Large N matrix models for 3d N = 2 theories: twisted index, free energy and black holes, JHEP 08 (2016) 064 [arXiv:1604.03122] [INSPIRE].
S.M. Hosseini and N. Mekareeya, Large N topologically twisted index: necklace quivers, dualities and Sasaki-Einstein spaces, JHEP 08 (2016) 089 [arXiv:1604.03397] [INSPIRE].
D. Martelli and J. Sparks, Moduli spaces of Chern-Simons quiver gauge theories and AdS4 /C F T3, Phys. Rev. D 78 (2008) 126005 [arXiv:0808.0912] [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].
D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [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].
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, H. Kim and N. Kim, Calculating the partition function of N = 2 Gauge theories on S3 and AdS/CFT correspondence, JHEP 05 (2011) 134 [arXiv:1102.5565] [INSPIRE].
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] [INSPIRE].
D.R. Gulotta, C.P. Herzog and S.S. Pufu, From necklace quivers to the F-theorem, operator counting and T (U(N)), JHEP 12 (2011) 077 [arXiv:1105.2817] [INSPIRE].
D.R. Gulotta, C.P. Herzog and S.S. Pufu, Operator counting and Eigenvalue distributions for 3D supersymmetric gauge theories, JHEP 11 (2011) 149 [arXiv:1106.5484] [INSPIRE].
S.M. Hosseini and A. Zaffaroni, Geometry of \( \mathcal{I} \) -extremization and black holes microstates, JHEP 07 (2019) 174 [arXiv:1904.04269] [INSPIRE].
J.P. Gauntlett, D. Martelli and J. Sparks, Toric geometry and the dual of \( \mathcal{I} \)-extremization, JHEP 06 (2019) 140 [arXiv:1904.04282] [INSPIRE].
D. Martelli, J. Sparks and S.-T. Yau, The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds, Commun. Math. Phys. 268 (2006) 39 [hep-th/0503183] [INSPIRE].
D. Fabbri et al., 3D superconformal theories from Sasakian seven manifolds: new nontrivial evidences for AdS4 /C F T3, Nucl. Phys. B 577 (2000) 547 [hep-th/9907219] [INSPIRE].
A. Hanany, D. Vegh and A. Zaffaroni, Brane tilings and M 2 branes, JHEP 03 (2009) 012 [arXiv:0809.1440] [INSPIRE].
J. Davey, A. Hanany, N. Mekareeya and G. Torri, Higgsing M 2-brane theories, JHEP 11 (2009) 028 [arXiv:0908.4033] [INSPIRE].
A. Amariti, C. Klare and M. Siani, The large N limit of toric Chern-Simons matter theories and their duals, JHEP 10 (2012) 019 [arXiv:1111.1723] [INSPIRE].
D. Martelli and J. Sparks, Notes on toric Sasaki-Einstein seven-manifolds and AdS4 /CFT3, JHEP 11 (2008) 016 [arXiv:0808.0904] [INSPIRE].
J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
N. Bobev, V.S. Min and K. Pilch, Mass-deformed ABJM and black holes in AdS4, JHEP 03 (2018) 050 [arXiv:1801.03135] [INSPIRE].
D. Cassani, P. Koerber and O. Varela, All homogeneous N = 2 M-theory truncations with supersymmetric AdS4 vacua, JHEP 11 (2012) 173 [arXiv:1208.1262] [INSPIRE].
J.P. Gauntlett, S. Kim, O. Varela and D. Waldram, Consistent supersymmetric Kaluza-Klein truncations with massive modes, JHEP 04 (2009) 102 [arXiv:0901.0676] [INSPIRE].
N. Halmagyi, M. Petrini and A. Zaffaroni, BPS black holes in AdS4 from M-theory, JHEP 08 (2013) 124 [arXiv:1305.0730] [INSPIRE].
F. Benini, H. Khachatryan and P. Milan, Black hole entropy in massive Type IIA, Class. Quant. Grav. 35 (2018) 035004 [arXiv:1707.06886] [INSPIRE].
S.M. Hosseini, K. Hristov and A. Passias, Holographic microstate counting for AdS4 black holes in massive IIA supergravity, JHEP 10 (2017) 190 [arXiv:1707.06884] [INSPIRE].
R. Monten and C. Toldo, Black holes with halos, Class. Quant. Grav. 35 (2018) 035001 [arXiv:1612.02399] [INSPIRE].
H. Kim and N. Kim, Operator counting for N = 2 Chern-Simons gauge theories with chiral-like matter fields, JHEP 05 (2012) 152 [arXiv:1202.6637] [INSPIRE].
H. Kim, N. Kim, S. Kim and J. Hun Lee, M-theory and seven-dimensional inhomogeneous Sasaki-Einstein manifolds, JHEP 01 (2011) 075 [arXiv:1011.4552] [INSPIRE].
J.P. Gauntlett, N. Kim and D. Waldram, Supersymmetric AdS3 , AdS2 and bubble solutions, JHEP 04 (2007) 005 [hep-th/0612253] [INSPIRE].
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ArXiv ePrint: 1904.05344
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Kim, H., Kim, N. Black holes with baryonic charge and \( \mathcal{I} \) -extremization. J. High Energ. Phys. 2019, 50 (2019). https://doi.org/10.1007/JHEP11(2019)050
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DOI: https://doi.org/10.1007/JHEP11(2019)050