##
**Geometry of \( \mathcal{I} \)-extremization and black holes microstates.**
*(English)*
Zbl 1418.83026

Summary: The entropy of a class of asymptotically \(\mathrm{AdS}_{4}\) magnetically charged BPS black holes can be obtained by extremizing the topologically twisted index of the dual three-dimensional field theory. This principle is known as \( \mathcal{I} \)-extremization. A gravitational dual of \( \mathcal{I} \)-extremization for a class of theories obtained by twisted compactifications of M2-branes living at a Calabi-Yau four-fold has been recently proposed. In this paper we investigate the relation between the two extremization principles. We show that the two extremization procedures are equivalent for theories without baryonic symmetries, which include ABJM and the theory dual to the non-toric Sasaki-Einstein manifold \(V^{5,2}\). We then consider a class of quivers dual to M2-branes at toric Calabi-Yau four-folds for which the \( \mathcal{I} \)-functional can be computed in the large N limit, and depends on three mesonic fluxes. We propose a gravitational dual for this construction, that we call mesonic twist, and we show that the gravitational extremization problem and \( \mathcal{I} \)-extremization are equivalent. We comment on more general cases.

### MSC:

83C57 | Black holes |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

53Z05 | Applications of differential geometry to physics |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

PDF
BibTeX
XML
Cite

\textit{S. M. Hosseini} and \textit{A. Zaffaroni}, J. High Energy Phys. 2019, No. 7, Paper No. 174, 52 p. (2019; Zbl 1418.83026)

### References:

[1] | D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP05 (2012) 159 [arXiv:1012.3210] [INSPIRE]. · Zbl 1348.81420 |

[2] | 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]. · Zbl 1190.53041 |

[3] | D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Commun. Math. Phys.280 (2008) 611 [hep-th/0603021] [INSPIRE]. · Zbl 1161.53029 |

[4] | 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]. |

[5] | 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, JHEP06 (2011) 102 [arXiv:1103.1181] [INSPIRE]. · Zbl 1298.81304 |

[6] | 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]. |

[7] | S. Cheon, H. Kim and N. Kim, Calculating the partition function of N = 2 gauge theories on S3and AdS/CFT correspondence, JHEP05 (2011) 134 [arXiv:1102.5565] [INSPIRE]. · Zbl 1296.81055 |

[8] | A. Amariti, C. Klare and M. Siani, The large N limit of toric Chern-Simons matter theories and their duals, JHEP10 (2012) 019 [arXiv:1111.1723] [INSPIRE]. · Zbl 1397.81323 |

[9] | P.M. Crichigno, C.P. Herzog and D. Jain, Free energy of Dnquiver Chern-Simons theories, JHEP03 (2013) 039 [arXiv:1211.1388] [INSPIRE]. · Zbl 1342.81571 |

[10] | A. Amariti, M. Fazzi, N. Mekareeya and A. Nedelin, New 3d N = 2 SCFT’s with N3/2scaling, arXiv:1903.02586 [INSPIRE]. · Zbl 1431.81141 |

[11] | F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS4from supersymmetric localization, JHEP05 (2016) 054 [arXiv:1511.04085] [INSPIRE]. · Zbl 1388.83388 |

[12] | 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]. · Zbl 1372.83030 |

[13] | F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP07 (2015) 127 [arXiv:1504.03698] [INSPIRE]. · Zbl 1388.81400 |

[14] | S.M. Hosseini and A. Zaffaroni, Large N matrix models for 3d N = 2 theories: twisted index, free energy and black holes, JHEP08 (2016) 064 [arXiv:1604.03122] [INSPIRE]. · Zbl 1390.83345 |

[15] | S.M. Hosseini and N. Mekareeya, Large N topologically twisted index: necklace quivers, dualities and Sasaki-Einstein spaces, JHEP08 (2016) 089 [arXiv:1604.03397] [INSPIRE]. · Zbl 1390.81600 |

[16] | A. Cabo-Bizet, V.I. Giraldo-Rivera and L.A. Pando Zayas, Microstate counting of AdS4hyperbolic black hole entropy via the topologically twisted index, JHEP08 (2017) 023 [arXiv:1701.07893] [INSPIRE]. · Zbl 1381.83055 |

[17] | F. Azzurli, N. Bobev, P.M. Crichigno, V.S. Min and A. Zaffaroni, A universal counting of black hole microstates in AdS4, JHEP02 (2018) 054 [arXiv:1707.04257] [INSPIRE]. · Zbl 1387.81303 |

[18] | S.M. Hosseini, K. Hristov and A. Passias, Holographic microstate counting for AdS4black holes in massive IIA supergravity, JHEP10 (2017) 190 [arXiv:1707.06884] [INSPIRE]. · Zbl 1383.83212 |

[19] | F. Benini, H. Khachatryan and P. Milan, Black hole entropy in massive type IIA, Class. Quant. Grav.35 (2018) 035004 [arXiv:1707.06886] [INSPIRE]. · Zbl 1382.83040 |

[20] | N. Bobev, V.S. Min and K. Pilch, Mass-deformed ABJM and black holes in AdS4, JHEP03 (2018) 050 [arXiv:1801.03135] [INSPIRE]. · Zbl 1388.83752 |

[21] | S.M. Hosseini, Black hole microstates and supersymmetric localization, Ph.D. thesis, Milan Bicocca U., Milan, Italy (2018) [arXiv:1803.01863] [INSPIRE]. |

[22] | S.M. Hosseini, A. Nedelin and A. Zaffaroni, The Cardy limit of the topologically twisted index and black strings in AdS5, JHEP04 (2017) 014 [arXiv:1611.09374] [INSPIRE]. · Zbl 1378.81125 |

[23] | J.T. Liu, L.A. Pando Zayas, V. Rathee and W. Zhao, One-loop test of quantum black holes in anti-de Sitter space, Phys. Rev. Lett.120 (2018) 221602 [arXiv:1711.01076] [INSPIRE]. |

[24] | J.T. Liu, L.A. Pando Zayas, V. Rathee and W. Zhao, Toward microstate counting beyond large N in localization and the dual one-loop quantum supergravity, JHEP01 (2018) 026 [arXiv:1707.04197] [INSPIRE]. · Zbl 1384.81111 |

[25] | I. Jeon and S. Lal, Logarithmic corrections to entropy of magnetically charged AdS4black holes, Phys. Lett.B 774 (2017) 41 [arXiv:1707.04208] [INSPIRE]. · Zbl 1403.83050 |

[26] | N. Halmagyi and S. Lal, On the on-shell: the action of AdS4black holes, JHEP03 (2018) 146 [arXiv:1710.09580] [INSPIRE]. · Zbl 1388.83662 |

[27] | A. Cabo-Bizet, U. Kol, L.A. Pando Zayas, I. Papadimitriou and V. Rathee, Entropy functional and the holographic attractor mechanism, JHEP05 (2018) 155 [arXiv:1712.01849] [INSPIRE]. · Zbl 1391.81150 |

[28] | C. Toldo and B. Willett, Partition functions on 3d circle bundles and their gravity duals, JHEP05 (2018) 116 [arXiv:1712.08861] [INSPIRE]. · Zbl 1391.83097 |

[29] | D. Gang and N. Kim, Large N twisted partition functions in 3d-3d correspondence and holography, Phys. Rev.D 99 (2019) 021901 [arXiv:1808.02797] [INSPIRE]. |

[30] | K. Hristov, I. Lodato and V. Reys, On the quantum entropy function in 4d gauged supergravity, JHEP07 (2018) 072 [arXiv:1803.05920] [INSPIRE]. · Zbl 1395.83126 |

[31] | J. Hong and J.T. Liu, The topologically twisted index of N = 4 super-Yang-Mills on T2× S2and the elliptic genus, JHEP07 (2018) 018 [arXiv:1804.04592] [INSPIRE]. · Zbl 1395.81268 |

[32] | J. Nian and X. Zhang, Entanglement entropy of ABJM theory and entropy of topological black hole, JHEP07 (2017) 096 [arXiv:1705.01896] [INSPIRE]. · Zbl 1380.83158 |

[33] | J.T. Liu, L.A. Pando Zayas and S. Zhou, Subleading microstate counting in the dual to massive type IIA, arXiv:1808.10445 [INSPIRE]. |

[34] | S.M. Hosseini, I. Yaakov and A. Zaffaroni, Topologically twisted indices in five dimensions and holography, JHEP11 (2018) 119 [arXiv:1808.06626] [INSPIRE]. · Zbl 1404.81232 |

[35] | P.M. Crichigno, D. Jain and B. Willett, 5d partition functions with a twist, JHEP11 (2018) 058 [arXiv:1808.06744] [INSPIRE]. · Zbl 1404.81223 |

[36] | S.M. Hosseini, K. Hristov, A. Passias and A. Zaffaroni, 6D attractors and black hole microstates, arXiv:1809.10685 [INSPIRE]. · Zbl 1405.83074 |

[37] | M. Suh, Supersymmetric AdS6black holes from matter coupled F(4) gauged supergravity, JHEP02 (2019) 108 [arXiv:1810.00675] [INSPIRE]. · Zbl 1411.83139 |

[38] | M. Suh, Supersymmetric AdS6black holes from F(4) gauged supergravity, JHEP01 (2019) 035 [arXiv:1809.03517] [INSPIRE]. · Zbl 1409.83214 |

[39] | D. Jain and A. Ray, 3d N = 2 ADÊ\[ \hat{ADE}\] Chern-Simons quivers, arXiv:1902.10498 [INSPIRE]. |

[40] | M. Fluder, S.M. Hosseini and C.F. Uhlemann, Black hole microstate counting in type IIB from 5d SCFTs, JHEP05 (2019) 134 [arXiv:1902.05074] [INSPIRE]. · Zbl 1416.83051 |

[41] | C. Couzens, J.P. Gauntlett, D. Martelli and J. Sparks, A geometric dual of c-extremization, JHEP01 (2019) 212 [arXiv:1810.11026] [INSPIRE]. · Zbl 1409.81143 |

[42] | J.P. Gauntlett, D. Martelli and J. Sparks, Toric geometry and the dual of c-extremization, JHEP01 (2019) 204 [arXiv:1812.05597] [INSPIRE]. · Zbl 1409.81118 |

[43] | S.M. Hosseini and A. Zaffaroni, Proving the equivalence of c-extremization and its gravitational dual for all toric quivers, JHEP03 (2019) 108 [arXiv:1901.05977] [INSPIRE]. · Zbl 1414.81208 |

[44] | O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP10 (2008) 091 [arXiv:0806.1218] [INSPIRE]. · Zbl 1245.81130 |

[45] | D. Martelli and J. Sparks, AdS4/CFT3duals from M2-branes at hypersurface singularities and their deformations, JHEP12 (2009) 017 [arXiv:0909.2036] [INSPIRE]. |

[46] | A. Amariti and M. Siani, Z extremization in chiral-like Chern Simons theories, JHEP06 (2012) 171 [arXiv:1109.4152] [INSPIRE]. · Zbl 1397.81127 |

[47] | D. Gang, C. Hwang, S. Kim and J. Park, Tests of AdS4/CFT3correspondence for N = 2 chiral-like theory, JHEP02 (2012) 079 [arXiv:1111.4529] [INSPIRE]. · Zbl 1309.81157 |

[48] | A. Donos and J.P. Gauntlett, Supersymmetric quantum criticality supported by baryonic charges, JHEP10 (2012) 120 [arXiv:1208.1494] [INSPIRE]. |

[49] | F. Benini, N. Bobev and P.M. Crichigno, Two-dimensional SCFTs from D3-branes, JHEP07 (2016) 020 [arXiv:1511.09462] [INSPIRE]. · Zbl 1390.83088 |

[50] | J.P. Gauntlett, D. Martelli and J. Sparks, Toric geometry and the dual ofI \[\mathcal{I} \]-extremization, JHEP06 (2019) 140 [arXiv:1904.04282] [INSPIRE]. · Zbl 1416.81155 |

[51] | 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]. · Zbl 0948.81619 |

[52] | S.S. Gubser and I.R. Klebanov, Baryons and domain walls in an N = 1 superconformal gauge theory, Phys. Rev.D 58 (1998) 125025 [hep-th/9808075] [INSPIRE]. |

[53] | D. Fabbri et al., 3D superconformal theories from Sasakian seven manifolds: new nontrivial evidences for AdS4/CFT3, Nucl. Phys.B 577 (2000) 547 [hep-th/9907219] [INSPIRE]. |

[54] | N. Benishti, D. Rodriguez-Gomez and J. Sparks, Baryonic symmetries and M5 branes in the AdS4/CFT3correspondence, JHEP07 (2010) 024 [arXiv:1004.2045] [INSPIRE]. · Zbl 1290.81101 |

[55] | O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP11 (2008) 043 [arXiv:0807.4924] [INSPIRE]. |

[56] | A. Hanany and A. Zaffaroni, Tilings, Chern-Simons theories and M2 branes, JHEP10 (2008) 111 [arXiv:0808.1244] [INSPIRE]. · Zbl 1245.81176 |

[57] | A. Hanany, D. Vegh and A. Zaffaroni, Brane tilings and M2 branes, JHEP03 (2009) 012 [arXiv:0809.1440] [INSPIRE]. |

[58] | D. Martelli and J. Sparks, Moduli spaces of Chern-Simons quiver gauge theories and AdS4/CFT3, Phys. Rev.D 78 (2008) 126005 [arXiv:0808.0912] [INSPIRE]. |

[59] | F. Benini, C. Closset and S. Cremonesi, Chiral flavors and M2-branes at toric CY4singularities, JHEP02 (2010) 036 [arXiv:0911.4127] [INSPIRE]. · Zbl 1270.81151 |

[60] | D.L. Jafferis, Quantum corrections to N = 2 Chern-Simons theories with flavor and their AdS4duals, JHEP08 (2013) 046 [arXiv:0911.4324] [INSPIRE]. · Zbl 1342.83381 |

[61] | A. Butti and A. Zaffaroni, R-charges from toric diagrams and the equivalence of a-maximization and Z-minimization, JHEP11 (2005) 019 [hep-th/0506232] [INSPIRE]. |

[62] | C. Couzens, D. Martelli and S. Schäfer-Nameki, F-theory and AdS3/CFT2(2, 0), JHEP06 (2018) 008 [arXiv:1712.07631] [INSPIRE]. · Zbl 1395.81211 |

[63] | M.M. Caldarelli and D. Klemm, Supersymmetry of anti-de Sitter black holes, Nucl. Phys.B 545 (1999) 434 [hep-th/9808097] [INSPIRE]. · Zbl 0953.83020 |

[64] | S. Cremonesi, N. Mekareeya and A. Zaffaroni, The moduli spaces of 3d N ≥ 2 Chern-Simons gauge theories and their Hilbert series, JHEP10 (2016) 046 [arXiv:1607.05728] [INSPIRE]. · Zbl 1390.81317 |

[65] | A. Amariti and S. Franco, Free energy vs Sasaki-Einstein volume for infinite families of M2-brane theories, JHEP09 (2012) 034 [arXiv:1204.6040] [INSPIRE]. · Zbl 1397.81210 |

[66] | K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, Nucl. Phys.B 667 (2003) 183 [hep-th/0304128] [INSPIRE]. · Zbl 1059.81602 |

[67] | F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP06 (2013) 005 [arXiv:1302.4451] [INSPIRE]. · Zbl 1390.83325 |

[68] | S. Cremonesi, Type IIB construction of flavoured ABJ(M) and fractional M2 branes, JHEP01 (2011) 076 [arXiv:1007.4562] [INSPIRE]. · Zbl 1214.81202 |

[69] | D.R. Morrison and M.R. Plesser, Nonspherical horizons. 1, Adv. Theor. Math. Phys.3 (1999) 1 [hep-th/9810201] [INSPIRE]. |

[70] | N. Halmagyi, M. Petrini and A. Zaffaroni, BPS black holes in AdS4from M-theory, JHEP08 (2013) 124 [arXiv:1305.0730] [INSPIRE]. · Zbl 1342.83168 |

[71] | S. Benvenuti, L.A. Pando Zayas and Y. Tachikawa, Triangle anomalies from Einstein manifolds, Adv. Theor. Math. Phys.10 (2006) 395 [hep-th/0601054] [INSPIRE]. · Zbl 1129.81081 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.