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Journal of High Energy Physics

, 2019:204 | Cite as

Toric geometry and the dual of c-extremization

  • Jerome P. GauntlettEmail author
  • Dario Martelli
  • James Sparks
Open Access
Regular Article - Theoretical Physics
  • 11 Downloads

Abstract

We consider D3-brane gauge theories at an arbitrary toric Calabi-Yau 3-fold cone singularity that are then further compactified on a Riemann surface Σg, with an arbitrary partial topological twist for the global U(1) symmetries. This constitutes a rich, infinite class of two-dimensional (0, 2) theories. Under the assumption that such a theory flows to a SCFT, we show that the supergravity formulas for the central charge and R-charges of BPS baryonic operators of the dual AdS3 solution may be computed using only the toric data of the Calabi-Yau 3-fold and the topological twist parameters. We exemplify the procedure for both the Yp,q and Xp,q 3-fold singularities, along with their associated dual quiver gauge theories, showing that the new supergravity results perfectly match the field theory results obtained using c-extremization, for arbitrary twist over Σg. We furthermore conjecture that the trial central charge Open image in new window , which we define in gravity, matches the field theory trial c-function off-shell, and show this holds in non-trivial examples. Finally, we check our general geometric formulae against a number of explicitly known supergravity solutions.

Keywords

AdS-CFT Correspondence 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]
    F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    C. Couzens, J.P. Gauntlett, D. Martelli and J. Sparks, A geometric dual of c-extremization, arXiv:1810.11026 [INSPIRE].
  4. [4]
    N. Kim, AdS 3 solutions of IIB supergravity from D3-branes, JHEP 01 (2006) 094 [hep-th/0511029] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    J.P. Gauntlett and N. Kim, Geometries with Killing spinors and supersymmetric AdS solutions, Commun. Math. Phys. 284 (2008) 897 [arXiv:0710.2590] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S. Franco, Y.-H. He, C. Sun and Y. Xiao, A comprehensive survey of brane tilings, Int. J. Mod. Phys. A 32 (2017) 1750142 [arXiv:1702.03958] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  7. [7]
    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
  8. [8]
    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
  9. [9]
    V. Guillemin, Kähler structures on toric varieties, J. Diff. Geom. 40 (1994) 285.CrossRefzbMATHGoogle Scholar
  10. [10]
    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
  11. [11]
    C. Couzens, D. Martelli and S. Schäfer-Nameki, F-theory and AdS 3 /CFT 2 (2, 0), JHEP 06 (2018) 008 [arXiv:1712.07631] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  12. [12]
    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.DG/0607586].
  13. [13]
    J.P. Gauntlett, D. Martelli, J. Sparks and S.-T. Yau, Obstructions to the existence of Sasaki-Einstein metrics, Commun. Math. Phys. 273 (2007) 803 [hep-th/0607080] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    T.C. Collins and G. Székelyhidi, Sasaki-Einstein metrics and K-stability, arXiv:1512.07213 [INSPIRE].
  15. [15]
    J.P. Gauntlett, O.A.P. Mac Conamhna, T. Mateos and D. Waldram, New supersymmetric AdS 3 solutions, Phys. Rev. D 74 (2006) 106007 [hep-th/0608055] [INSPIRE].ADSGoogle Scholar
  16. [16]
    F. Benini, N. Bobev and P.M. Crichigno, Two-dimensional SCFTs from D3-branes, JHEP 07 (2016) 020 [arXiv:1511.09462] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    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
  19. [19]
    A. Donos, J.P. Gauntlett and N. Kim, AdS solutions through transgression, JHEP 09 (2008) 021 [arXiv:0807.4375] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    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
  21. [21]
    J.P. Gauntlett, O.A.P. Mac Conamhna, T. Mateos and D. Waldram, Supersymmetric AdS 3 solutions of type IIB supergravity, Phys. Rev. Lett. 97 (2006) 171601 [hep-th/0606221] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    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
  23. [23]
    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].ADSCrossRefGoogle Scholar
  24. [24]
    K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    J.P. Gauntlett, N. Kim and D. Waldram, Supersymmetric AdS 3 , AdS 2 and bubble solutions, JHEP 04 (2007) 005 [hep-th/0612253] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    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].ADSCrossRefzbMATHGoogle Scholar
  27. [27]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    J.P. Gauntlett, N. Kim, S. Pakis and D. Waldram, Membranes wrapped on holomorphic curves, Phys. Rev. D 65 (2002) 026003 [hep-th/0105250] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    A. Donos, J.P. Gauntlett and C. Pantelidou, Magnetic and electric AdS solutions in string- and M-theory, Class. Quant. Grav. 29 (2012) 194006 [arXiv:1112.4195] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    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
  31. [31]
    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
  32. [32]
    A. Amariti, L. Cassia and S. Penati, c-extremization from toric geometry, Nucl. Phys. B 929 (2018) 137 [arXiv:1706.07752] [INSPIRE].
  33. [33]
    D. Forcella, A. Hanany, Y.-H. He and A. Zaffaroni, The master space of N = 1 gauge theories, JHEP 08 (2008) 012 [arXiv:0801.1585] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Jerome P. Gauntlett
    • 1
    Email author
  • Dario Martelli
    • 2
  • James Sparks
    • 3
  1. 1.Blackett LaboratoryImperial CollegeLondonU.K.
  2. 2.Department of MathematicsKing’s College LondonLondonU.K.
  3. 3.Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory QuarterOxfordU.K.

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