Advertisement

Journal of High Energy Physics

, 2019:252 | Cite as

Localization of the action in AdS/CFT

  • Pietro Benetti GenoliniEmail author
  • Juan Manuel Perez Ipiña
  • James Sparks
Open Access
Regular Article - Theoretical Physics
  • 26 Downloads

Abstract

We derive a simple formula for the action of any supersymmetric solution to minimal gauged supergravity in the AdS4/CFT3 correspondence. Such solutions are equipped with a supersymmetric Killing vector, and we show that the holographically renormalized action may be expressed entirely in terms of the weights of this vector field at its fixed points, together with certain topological data. In this sense, the classical gravitational partition function localizes in the bulk. We illustrate our general formula with a number of explicit examples, in which exact dual field theory computations are also available, which include supersymmetric Taub-NUT and Taub-bolt type spacetimes, as well as black hole solutions. Our simple topological formula also allows us to write down the action of any solution, provided it exists.

Keywords

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

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

References

  1. [1]
    V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
  2. [2]
    A. Dabholkar, N. Drukker and J. Gomes, Localization in supergravity and quantum AdS 4 /C F T 3 holography, JHEP 10 (2014) 090 [arXiv:1406.0505] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    K. Hristov, I. Lodato and V. Reys, On the quantum entropy function in 4d gauged supergravity, JHEP 07 (2018) 072 [arXiv:1803.05920] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    B. de Wit, S. Murthy and V. Reys, BRST quantization and equivariant cohomology: localization with asymptotic boundaries, JHEP 09 (2018) 084 [arXiv:1806.03690] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  5. [5]
    I. Jeon and S. Murthy, Twisting and localization in supergravity: equivariant cohomology of BPS black holes, JHEP 03 (2019) 140 [arXiv:1806.04479] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  7. [7]
    D.Z. Freedman and A.K. Das, Gauge internal symmetry in extended supergravity, Nucl. Phys. B 120 (1977) 221 [INSPIRE].
  8. [8]
    J.P. Gauntlett and O. Varela, Consistent Kaluza-Klein reductions for general supersymmetric AdS solutions, Phys. Rev. D 76 (2007) 126007 [arXiv:0707.2315] [INSPIRE].
  9. [9]
    D. Martelli, A. Passias and J. Sparks, The gravity dual of supersymmetric gauge theories on a squashed three-sphere, Nucl. Phys. B 864 (2012) 840 [arXiv:1110.6400] [INSPIRE].
  10. [10]
    D. Martelli and J. Sparks, The gravity dual of supersymmetric gauge theories on a biaxially squashed three-sphere, Nucl. Phys. B 866 (2013) 72 [arXiv:1111.6930] [INSPIRE].
  11. [11]
    D. Martelli, A. Passias and J. Sparks, The supersymmetric NUTs and bolts of holography, Nucl. Phys. B 876 (2013) 810 [arXiv:1212.4618] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    D. Martelli and A. Passias, The gravity dual of supersymmetric gauge theories on a two-parameter deformed three-sphere, Nucl. Phys. B 877 (2013) 51 [arXiv:1306.3893] [INSPIRE].
  13. [13]
    D. Farquet, J. Lorenzen, D. Martelli and J. Sparks, Gravity duals of supersymmetric gauge theories on three-manifolds, JHEP 08 (2016) 080 [arXiv:1404.0268] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    F. Azzurli et al., A universal counting of black hole microstates in AdS 4 , JHEP 02 (2018) 054 [arXiv:1707.04257] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    C. Toldo and B. Willett, Partition functions on 3d circle bundles and their gravity duals, JHEP 05 (2018) 116 [arXiv:1712.08861] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric Field Theories on Three-Manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  19. [19]
    G.W. Gibbons and S.W. Hawking, Classification of gravitational instanton symmetries, Commun. Math. Phys. 66 (1979) 291 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    M. Dunajski, J.B. Gutowski, W.A. Sabra and P. Tod, Cosmological Einstein-Maxwell instantons and Euclidean supersymmetry: beyond self-duality, JHEP 03 (2011) 131 [arXiv:1012.1326] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    P. Benetti Genolini, D. Cassani, D. Martelli and J. Sparks, Holographic renormalization and supersymmetry, JHEP 02 (2017) 132 [arXiv:1612.06761] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    M. Dunajski, J. Gutowski, W. Sabra and P. Tod, Cosmological Einstein-Maxwell instantons and Euclidean supersymmetry: anti-self-dual solutions, Class. Quant. Grav. 28 (2011) 025007 [arXiv:1006.5149] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Large N phases, gravitational instantons and the nuts and bolts of AdS holography, Phys. Rev. D 59 (1999) 064010 [hep-th/9808177] [INSPIRE].
  24. [24]
    P. Orlik and F. Raymond, Actions of the torus on 4-manifolds. I, Trans. AMer. Math. Soc. 152 (1970) 531.MathSciNetzbMATHGoogle Scholar
  25. [25]
    D.M.J. Calderbank and M.A. Singer, Einstein metrics and complex singularities, Invent. Math. 156 (2004) 405 [math/0206229].
  26. [26]
    L.J. Romans, Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory, Nucl. Phys. B 383 (1992) 395 [hep-th/9203018] [INSPIRE].
  27. [27]
    D.R. Brill, J. Louko and P. Peldan, Thermodynamics of (3 + 1)-dimensional black holes with toroidal or higher genus horizons, Phys. Rev. D 56 (1997) 3600 [gr-qc/9705012] [INSPIRE].
  28. [28]
    M.M. Caldarelli and D. Klemm, Supersymmetry of Anti-de Sitter black holes, Nucl. Phys. B 545 (1999) 434 [hep-th/9808097] [INSPIRE].
  29. [29]
    A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS 5 black holes, arXiv:1810.11442 [INSPIRE].
  30. [30]
    D. Cassani and L. Papini, The BPS limit of rotating AdS black hole thermodynamics, JHEP 09 (2019) 079 [arXiv:1906.10148] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS 4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    F. Benini, N. Bobev and P.M. Crichigno, Two-dimensional SCFTs from D3-branes, JHEP 07 (2016) 020 [arXiv:1511.09462] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    N. Bobev and P.M. Crichigno, Universal RG flows across dimensions and holography, JHEP 12 (2017) 065 [arXiv:1708.05052] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    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].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    C. Closset, H. Kim and B. Willett, Supersymmetric partition functions and the three-dimensional A-twist, JHEP 03 (2017) 074 [arXiv:1701.03171] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    C. Closset, H. Kim and B. Willett, Seifert fibering operators in 3d \( \mathcal{N} \) = 2 theories, JHEP 11 (2018) 004 [arXiv:1807.02328] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    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].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    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].
  39. [39]
    D.M.J. Calderbank and H. Pedersen, Selfdual Einstein metrics with torus symmetry, J. Diff. Geom. 60 (2002) 485 [math/0105263] [INSPIRE].
  40. [40]
    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].ADSCrossRefGoogle Scholar
  41. [41]
    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, JHEP 01 (2018) 026 [arXiv:1707.04197] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    J.T. Liu, L.A. Pando Zayas and S. Zhou, Subleading microstate counting in the dual to massive Type IIA, arXiv:1808.10445 [INSPIRE].
  43. [43]
    D. Gang, N. Kim and L.A. Pando Zayas, Precision microstate counting for the entropy of wrapped M 5-branes, arXiv:1905.01559 [INSPIRE].
  44. [44]
    N. Halmagyi and S. Lal, On the on-shell: the action of AdS4 black holes, JHEP 03 (2018) 146 [arXiv:1710.09580] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    A. Cabo-Bizet et al., Entropy functional and the holographic attractor mechanism, JHEP 05 (2018) 155 [arXiv:1712.01849] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    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].ADSCrossRefGoogle Scholar
  47. [47]
    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].ADSMathSciNetCrossRefGoogle 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]
    F. Benini, K. Hristov and A. Zaffaroni, Exact microstate counting for dyonic black holes in AdS 4 , Phys. Lett. B 771 (2017) 462 [arXiv:1608.07294] [INSPIRE].
  50. [50]
    A. Cabo-Bizet, V.I. Giraldo-Rivera and L.A. Pando Zayas, Microstate counting of AdS 4 hyperbolic black hole entropy via the topologically twisted index, JHEP 08 (2017) 023 [arXiv:1701.07893] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    S.M. Hosseini, K. Hristov and A. Passias, Holographic microstate counting for AdS 4 black holes in massive IIA supergravity, JHEP 10 (2017) 190 [arXiv:1707.06884] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    F. Benini, H. Khachatryan and P. Milan, Black hole entropy in massive Type IIA, Class. Quant. Grav. 35 (2018) 035004 [arXiv:1707.06886] [INSPIRE].
  53. [53]
    N. Bobev, V.S. Min and K. Pilch, Mass-deformed ABJM and black holes in AdS 4, JHEP 03 (2018) 050 [arXiv:1801.03135] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    K. Hristov, S. Katmadas and C. Toldo, Rotating attractors and BPS black holes in AdS 4, JHEP 01 (2019) 199 [arXiv:1811.00292] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeU.K.
  2. 2.Mathematical InstituteUniversity of OxfordOxfordU.K.

Personalised recommendations