Proving the equivalence of c-extremization and its gravitational dual for all toric quivers

  • Seyed Morteza HosseiniEmail author
  • Alberto Zaffaroni
Open Access
Regular Article - Theoretical Physics


The gravitational dual of c-extremization for a class of (0, 2) two-dimensional theories obtained by twisted compactifications of D3-brane gauge theories living at a toric Calabi-Yau three-fold has been recently proposed. The equivalence of this construction with c-extremization has been checked in various examples and holds also off-shell. In this note we prove that such equivalence holds for an arbitrary toric Calabi-Yau. We do it by generalizing the proof of the equivalence between a-maximization and volume minimization for four-dimensional toric quivers. By an explicit parameterization of the R-charges we map the trial right-moving central charge cr into the off-shell functional to be extremized in gravity. We also observe that the similar construction for M2-branes on ℂ4 is equivalent to the ℐ-extremization principle that leads to the microscopic counting for the entropy of magnetically charged black holes in AdS4 × S7. Also this equivalence holds off-shell.


AdS-CFT Correspondence Black Holes in String Theory Supersymmetric Gauge Theory 


Open Access

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  1. [1]
    K.A. Intriligator and B. Wecht, The Exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
  2. [2]
    F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [INSPIRE].
  3. [3]
    F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  4. [4]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Commun. Math. Phys. 280 (2008) 611 [hep-th/0603021] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Y. Tachikawa, Five-dimensional supergravity dual of a-maximization, Nucl. Phys. B 733 (2006) 188 [hep-th/0507057] [INSPIRE].
  7. [7]
    P. Szepietowski, Comments on a-maximization from gauged supergravity, JHEP 12 (2012) 018 [arXiv:1209.3025] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    P. Karndumri and E. O Colgain, Supergravity dual of c-extremization, Phys. Rev. D 87 (2013) 101902 [arXiv:1302.6532] [INSPIRE].
  9. [9]
    A. Butti and A. Zaffaroni, R-charges from toric diagrams and the equivalence of a-maximization and Z-minimization, JHEP 11 (2005) 019 [hep-th/0506232] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    S. Lee and S.-J. Rey, Comments on anomalies and charges of toric-quiver duals, JHEP 03 (2006) 068 [hep-th/0601223] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R. Eager, Equivalence of A-Maximization and Volume Minimization, JHEP 01 (2014) 089 [arXiv:1011.1809] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    C. Couzens, J.P. Gauntlett, D. Martelli and J. Sparks, A geometric dual of c-extremization, JHEP 01 (2019) 212 [arXiv:1810.11026] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  13. [13]
    J.P. Gauntlett, D. Martelli and J. Sparks, Toric geometry and the dual of c-extremization, JHEP 01 (2019) 204 [arXiv:1812.05597] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [INSPIRE].
  15. [15]
    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].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    B. Feng, Y.-H. He, K.D. Kennaway and C. Vafa, Dimer models from mirror symmetry and quivering amoebae, Adv. Theor. Math. Phys. 12 (2008) 489 [hep-th/0511287] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    S.M. Hosseini, A. Nedelin and A. Zaffaroni, The Cardy limit of the topologically twisted index and black strings in AdS 5, JHEP 04 (2017) 014 [arXiv:1611.09374] [INSPIRE].
  18. [18]
    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].
  19. [19]
    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].
  20. [20]
    S.M. Hosseini and A. Zaffaroni, Large N matrix models for 3d \( \mathcal{N}=2 \) theories: twisted index, free energy and black holes, JHEP 08 (2016) 064 [arXiv:1604.03122] [INSPIRE].
  21. [21]
    F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS 4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
  22. [22]
    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].
  23. [23]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    F. Azzurli, N. Bobev, P.M. Crichigno, V.S. Min and A. Zaffaroni, A universal counting of black hole microstates in AdS 4, JHEP 02 (2018) 054 [arXiv:1707.04257] [INSPIRE].
  25. [25]
    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].
  26. [26]
    B.S. Acharya, J.M. Figueroa-O’Farrill, C.M. Hull and B.J. Spence, Branes at conical singularities and holography, Adv. Theor. Math. Phys. 2 (1999) 1249 [hep-th/9808014] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    D.R. Morrison and M.R. Plesser, Nonspherical horizons. 1., Adv. Theor. Math. Phys. 3 (1999) 1 [hep-th/9810201] [INSPIRE].
  28. [28]
    J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Sasaki-Einstein metrics on S 2 × S 3, Adv. Theor. Math. Phys. 8 (2004) 711 [hep-th/0403002] [INSPIRE].
  29. [29]
    J.P. Gauntlett, D. Martelli, J.F. Sparks and D. Waldram, A New infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys. 8 (2004) 987 [hep-th/0403038] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Cvetič, H. Lü, D.N. Page and C.N. Pope, New Einstein-Sasaki spaces in five and higher dimensions, Phys. Rev. Lett. 95 (2005) 071101 [hep-th/0504225] [INSPIRE].
  31. [31]
    S. Benvenuti, S. Franco, A. Hanany, D. Martelli and J. Sparks, An Infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals, JHEP 06 (2005) 064 [hep-th/0411264] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    S. Benvenuti and M. Kruczenski, From Sasaki-Einstein spaces to quivers via BPS geodesics: L p,q|r , JHEP 04 (2006) 033 [hep-th/0505206] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    A. Butti, D. Forcella and A. Zaffaroni, The Dual superconformal theory for L p,q,r manifolds, JHEP 09 (2005) 018 [hep-th/0505220] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht, Gauge theories from toric geometry and brane tilings, JHEP 01 (2006) 128 [hep-th/0505211] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    A. Butti and A. Zaffaroni, From toric geometry to quiver gauge theory: The Equivalence of a-maximization and Z-minimization, Fortsch. Phys. 54 (2006) 309 [hep-th/0512240] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    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].
  39. [39]
    S.S. Gubser, Einstein manifolds and conformal field theories, Phys. Rev. D 59 (1999) 025006 [hep-th/9807164] [INSPIRE].
  40. [40]
    A. Amariti, L. Cassia and S. Penati, c-extremization from toric geometry, Nucl. Phys. B 929 (2018) 137 [arXiv:1706.07752] [INSPIRE].
  41. [41]
    S.M. Hosseini, I. Yaakov and A. Zaffaroni, Topologically twisted indices in five dimensions and holography, JHEP 11 (2018) 119 [arXiv:1808.06626] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  42. [42]
    F. Benini, N. Bobev and P.M. Crichigno, Two-dimensional SCFTs from D3-branes, JHEP 07 (2016) 020 [arXiv:1511.09462] [INSPIRE].
  43. [43]
    A. Amariti and C. Toldo, Betti multiplets, flows across dimensions and c-extremization, JHEP 07 (2017) 040 [arXiv:1610.08858] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    A. Amariti, L. Cassia and S. Penati, Surveying 4d SCFTs twisted on Riemann surfaces, JHEP 06 (2017) 056 [arXiv:1703.08201] [INSPIRE].
  45. [45]
    C. Couzens, D. Martelli and S. Schäfer-Nameki, F-theory and AdS 3 /CF T 2 (2, 0), JHEP 06 (2018) 008 [arXiv:1712.07631] [INSPIRE].
  46. [46]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  49. [49]
    C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP 08 (2016) 059 [arXiv:1605.06531] [INSPIRE].
  50. [50]
    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].
  51. [51]
    S.L. Cacciatori and D. Klemm, Supersymmetric AdS 4 black holes and attractors, JHEP 01 (2010) 085 [arXiv:0911.4926] [INSPIRE].
  52. [52]
    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].
  53. [53]
    K. Hristov and S. Vandoren, Static supersymmetric black holes in AdS 4 with spherical symmetry, JHEP 04 (2011) 047 [arXiv:1012.4314] [INSPIRE].
  54. [54]
    A. Hanany, D. Vegh and A. Zaffaroni, Brane Tilings and M2 Branes, JHEP 03 (2009) 012 [arXiv:0809.1440] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, Phys. Rev. Lett. 105 (2010) 141601 [arXiv:0909.4776] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  56. [56]
    I. Bah, C. Beem, N. Bobev and B. Wecht, AdS/CFT Dual Pairs from M5-Branes on Riemann Surfaces, Phys. Rev. D 85 (2012) 121901 [arXiv:1112.5487] [INSPIRE].
  57. [57]
    I. Bah, C. Beem, N. Bobev and B. Wecht, Four-Dimensional SCFTs from M5-Branes, JHEP 06 (2012) 005 [arXiv:1203.0303] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Kavli IPMU (WPI), UTIAS, The University of TokyoKashiwaJapan
  2. 2.Dipartimento di FisicaUniversità di Milano-BicoccaMilanoItaly
  3. 3.INFN, sezione di Milano-BicoccaMilanoItaly

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