Holography, matrix factorizations and K-stability

Abstract

Placing D3-branes at conical Calabi-Yau threefold singularities produces many AdS5/CFT4 duals. Recent progress in differential geometry has produced a technique (called K-stability) to recognize which singularities admit conical Calabi-Yau metrics. On the other hand, the algebraic technique of non-commutative crepant resolutions, involving matrix factorizations, has been developed to associate a quiver to a singularity. In this paper, we put together these ideas to produce new AdS5/CFT4 duals, with special emphasis on non-toric singularities.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    A. Futaki, H. Ono and G. Wang, Transverse Kahler geometry of Sasaki manifolds and toric Sasaki- Einstein manifolds, J. Diff. Geom. 83 (2009) 585 [math . DG/0607586] [INSPIRE].

  3. [3]

    J. Sparks, Sasaki-Einstein manifolds, Surveys Diff. Geom.16 (2011) 265 [arXiv :1004 . 2461] [INSPIRE].

  4. [4]

    T.C. Collins and G. Szekelyhidi, Sasaki-Einstein metrics and K-stability, Geom. Topol.23 (2019) 1339 [arXiv:1512 .07213] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    X.-X. Chen, S. Donaldson and S. Sun, Kähler-Einstein metrics and stability, arXiv:1210.7494.

  6. [6]

    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].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  7. [7]

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. [8]

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

    ADS  MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    K. Altmann and J. Hausen, Polyhedral divisors and algebraic torus actions, Math. Annalen334 (2006) 557 [math.AG/0306285].

  10. [10]

    N. Ilten and H. Süß, K-stability for Fano manifolds with torus action of complexity one, Duke Math. J.166 (2017) 177 [arXiv:1507.04442].

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    T.C. Collins, D. Xie and S.-T. Yau, K-stability and stability of chiral ring, arXiv:1606.09260 [INSPIRE].

  12. [12]

    A. Hanany and K.D. Kennaway, Dimer models and toric diagrams, hep-th/0503149 [INSPIRE].

  13. [13]

    S. Franco, A. Hanany, K.D. Kennaway, D. Vegh and B. Wecht, Brane dimers and quiver gauge theories, JHEP01 (2006) 096 [hep-th/0504110] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  14. [14]

    M. van den Bergh, Non-commutative crepant resolutions, in The legacy of Niels Henrik Abel, Springer, Berlin, Germany (2004), pg. 749.

  15. [15]

    P.S. Aspinwall and D.R. Morrison, Quivers from matrix factorizations, Commun. Math. Phys.313 (2012) 607 [arXiv:1005.1042] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  16. [16]

    M. Wemyss, Lectures on noncommutative resolutions, arXiv:1210.2564 [INSPIRE].

  17. [17]

    O. Iyama and M. Wemyss, Reduction of triangulated categories and maximal modification algebras for cAnsingularities, J. Reine Angew. Math.738 (2018) 149.

    MathSciNet  MATH  Article  Google Scholar 

  18. [18]

    S. Gubser, N. Nekrasov and S. Shatashvili, Generalized conifolds and 4-dimensional N = 1 superconformal field theory, JHEP05 (1999) 003 [hep-th/9811230] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  19. [19]

    A. Amariti, M. Fazzi, N. Mekareeya and A. Nedelin, New 3d N = 2 SCFT’s with N3/2scaling, JHEP12 (2019) 111 [arXiv:1903.02586] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  20. [20]

    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  21. [21]

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

  22. [22]

    J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Sasaki-Einstein metrics on S2× S3 , Adv. Theor. Math. Phys.8 (2004) 711 [hep-th/0403002] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  23. [23]

    A. Bergman and C.P. Herzog, The volume of some nonspherical horizons and the AdS/CFT correspondence, JHEP01 (2002) 030 [hep-th/0108020] [INSPIRE].

    ADS  Article  Google Scholar 

  24. [24]

    R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory, Nucl. Phys.B 447 (1995) 95 [hep-th/9503121] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  25. [25]

    D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys.B 526 (1998) 543 [hep-th/9708042] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  26. [26]

    S. Benvenuti and A. Hanany, New results on superconformal quivers, JHEP04 (2006) 032 [hep-th/0411262] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. [27]

    S.S. Gubser, Einstein manifolds and conformal field theories, Phys. Rev.D 59 (1999) 025006 [hep-th/9807164] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  28. [28]

    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP07 (1998) 023 [hep-th/9806087] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  29. [29]

    T.C. Collins, Canonical metrics in Sasakian geometry, ProQuest LLC, Ann Arbor, MI, U.S.A. (2014).

  30. [30]

    S.K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc.s3-50 (1985) 1.

  31. [31]

    K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math.39 (1986) S257.

    MathSciNet  MATH  Article  Google Scholar 

  32. [32]

    S.-T. Yau, Review on Kähler-Einstein metrics in algebraic geometry, in Proceedings of the Hirzebruch 65thconference on algebraic geometry (Ramat Gan, 1993), in Israel Math. Conf. Proc.9, Bar-Ilan Univ., Israel (1996), pg. 433.

  33. [33]

    J. Stoppa, K-stability of constant scalar curvature Kähler manifolds, Adv. Math.221 (2009) 1397 [arXiv:0803.4095].

    MathSciNet  MATH  Article  Google Scholar 

  34. [34]

    S. Benvenuti and S. Giacomelli, Supersymmetric gauge theories with decoupled operators and chiral ring stability, Phys. Rev. Lett.119 (2017) 251601 [arXiv:1706.02225] [INSPIRE].

    ADS  Article  Google Scholar 

  35. [35]

    A. Liendo and H. Süß, Normal singularities with torus actions, Tohoku Math. J.65 (2013) 105 [arXiv:1005.2462].

    MathSciNet  MATH  Article  Google Scholar 

  36. [36]

    S.-T. Yau and Y. Yu, Classification of 3-dimensional isolated rational hypersurface singularities with C-action, Rocky Mountain J. Math.35 (2005) 1795 [math.AG/0303302] [INSPIRE].

  37. [37]

    M. Futaki and K. Ueda, Homological mirror symmetry for Brieskorn-Pham singularities, arXiv:0912.0316.

  38. [38]

    D. Berenstein and R.G. Leigh, Resolution of stringy singularities by noncommutative algebras, JHEP06 (2001) 030 [hep-th/0105229] [INSPIRE].

    ADS  Article  Google Scholar 

  39. [39]

    G.J. Leuschke and R. Wiegand, Cohen-Macaulay representations, Math. Surv. Monogr.181, American Mathematical Society, Providence, RI, U.S.A. (2012).

  40. [40]

    D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc.260 (1980) 35.

    MathSciNet  MATH  Article  Google Scholar 

  41. [41]

    A. Kapustin and Y. Li, D-branes in Landau-Ginzburg models and algebraic geometry, JHEP12 (2003) 005 [hep-th/0210296] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  42. [42]

    M. Baumgartl, I. Brunner and M.R. Gaberdiel, D-brane superpotentials and RG flows on the quintic, JHEP07 (2007) 061 [arXiv:0704.2666] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  43. [43]

    I. Brunner, M. Herbst, W. Lerche and J. Walcher, Matrix factorizations and mirror symmetry: the cubic curve, JHEP11 (2006) 006 [hep-th/0408243] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  44. [44]

    K. Hori and J. Walcher, D-branes from matrix factorizations, Comptes Rendus Physique5 (2004) 1061 [hep-th/0409204] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  45. [45]

    S. Gukov and J. Walcher, Matrix factorizations and Kauffman homology, hep-th/0512298 [INSPIRE].

  46. [46]

    I. Brunner, M. Herbst, W. Lerche and B. Scheuner, Landau-Ginzburg realization of open string TFT, JHEP11 (2006) 043 [hep-th/0305133] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  47. [47]

    I. Brunner, M.R. Gaberdiel and C.A. Keller, Matrix factorisations and D-branes on K 3, JHEP06 (2006) 015 [hep-th/0603196] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  48. [48]

    A. Caldararu, J. Distler, S. Hellerman, T. Pantev and E. Sharpe, Non-birational twisted derived equivalences in Abelian GLSMs, Commun. Math. Phys.294 (2010) 605 [arXiv:0709.3855] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  49. [49]

    M. Herbst, K. Hori and D. Page, Phases of N = 2 theories in 1 + 1 dimensions with boundary, arXiv:0803.2045 [INSPIRE].

  50. [50]

    A.P. Braun, A. Collinucci and R. Valandro, G-flux in F-theory and algebraic cycles, Nucl. Phys.B 856 (2012) 129 [arXiv:1107.5337] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  51. [51]

    N.M. Addington, E.P. Segal and E. Sharpe, D-brane probes, branched double covers and noncommutative resolutions, Adv. Theor. Math. Phys.18 (2014) 1369 [arXiv:1211.2446] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  52. [52]

    E. Sharpe, Predictions for Gromov-Witten invariants of noncommutative resolutions, J. Geom. Phys.74 (2013) 256 [arXiv:1212.5322] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  53. [53]

    A. Collinucci and R. Savelli, F-theory on singular spaces, JHEP09 (2015) 100 [arXiv:1410.4867] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  54. [54]

    A. Collinucci and R. Savelli, T-branes as branes within branes, JHEP09 (2015) 161 [arXiv:1410.4178] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  55. [55]

    A. Collinucci, M. Fazzi and R. Valandro, Geometric engineering on flops of length two, JHEP04 (2018) 090 [arXiv:1802.00813] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  56. [56]

    A. Collinucci, M. Fazzi, D.R. Morrison and R. Valandro, High electric charges in M-theory from quiver varieties, JHEP11 (2019) 111 [arXiv:1906.02202] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  57. [57]

    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].

    ADS  MATH  Article  Google Scholar 

  58. [58]

    M.R. Douglas, D-branes, categories and N = 1 supersymmetry, J. Math. Phys.42 (2001) 2818 [hep-th/0011017] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  59. [59]

    C. Beil, On the noncommutative geometry of square superpotential algebras, J. Algebra371 (2012) 207 [arXiv:0811.2439] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  60. [60]

    Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Math. Soc. Lect. Note Ser.146, Cambridge University Press, Cambridge, U.K. (1990).

  61. [61]

    M. Reid, Minimal models of canonical 3-folds, in Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math.1, North-Holland, Amsterdam, The Netherlands (1983), pg. 1.

  62. [62]

    I. Burban, O. Iyama, B. Keller and I. Reiten, Cluster tilting for one-dimensional hypersurface singularities, arXiv:0704.1249.

  63. [63]

    R. Corrado and N. Halmagyi, N = 1 field theories and fluxes in IIB string theory, Phys. Rev.D 71 (2005) 046001 [hep-th/0401141] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  64. [64]

    P.S. Aspinwall and S.H. Katz, Computation of superpotentials for D-branes, Commun. Math. Phys.264 (2006) 227 [hep-th/0412209] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  65. [65]

    F. Cachazo, B. Fiol, K.A. Intriligator, S. Katz and C. Vafa, A geometric unification of dualities, Nucl. Phys.B 628 (2002) 3 [hep-th/0110028] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  66. [66]

    M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].

  67. [67]

    A. Butti, A. Zaffaroni and D. Forcella, Deformations of conformal theories and non-toric quiver gauge theories, JHEP02 (2007) 081 [hep-th/0607147] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  68. [68]

    E. Lopez, A family of N = 1 SU(N )ktheories from branes at singularities, JHEP02 (1999) 019 [hep-th/9812025] [INSPIRE].

    ADS  Article  Google Scholar 

  69. [69]

    A. Némethi, “Weakly” elliptic Gorenstein singularities of surfaces, Invent. Math.137 (1999) 145 [math.AG/9808119].

  70. [70]

    P. Wagreich, Elliptic singularities of surfaces, Amer. J. Math.92 (1970) 419.

    MathSciNet  MATH  Article  Google Scholar 

  71. [71]

    S.-T. Yau, Normal two-dimensional elliptic singularities, ProQuest LLC, Ann Arbor, MI, U.S.A. (1976).

  72. [72]

    S.-T. Yau, On almost minimally elliptic singularities, Bull. Amer. Math. Soc.83 (1977) 362.

    MathSciNet  MATH  Article  Google Scholar 

  73. [73]

    H.B. Laufer, On minimally elliptic singularities, Amer. J. Math.99 (1977) 1257.

    MathSciNet  MATH  Article  Google Scholar 

  74. [74]

    M. Artin, On isolated rational singularities of surfaces, Amer. J. Math.88 (1966) 129.

    MathSciNet  MATH  Article  Google Scholar 

  75. [75]

    C.P. Kahn, Reflexive modules on minimally elliptic singularities, Math. Annalen285 (1989) 141.

    MathSciNet  MATH  Article  Google Scholar 

  76. [76]

    Y. Drozd, G.-M. Greuel and I. Kashuba, On Cohen-Macaulay modules on surface singularities, Moscow Math. J.3 (2003) 397.

    MathSciNet  MATH  Article  Google Scholar 

  77. [77]

    V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of differentiable maps. Volume 1, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, NY, U.S.A. (2012).

  78. [78]

    N. Kasuya, On the links of simple singularities, simple elliptic singularities and cusp singularities, Demonstr. Math.48 (2015) 289.

    MathSciNet  MATH  Google Scholar 

  79. [79]

    C.P.M. Kahn, Reflexive Moduln auf einfach-elliptischen Flächensingularitäten (in Germany), Bonner Mathematische Schriften [Bonn Mathematical Publications]188, Mathematisches Institut, Universität Bonn, Bonn, Germany (1988).

  80. [80]

    M. Wijnholt, Large volume perspective on branes at singularities, Adv. Theor. Math. Phys.7 (2003) 1117 [hep-th/0212021] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  81. [81]

    W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-1-1 — a computer algebra system for polynomial computations, http://www.singular.uni-kl.de, (2018).

  82. [82]

    M. Wemyss, Aspects of the homological minimal model program, arXiv:1411.7189.

  83. [83]

    J. Karmazyn, The length classification of threefold flops via noncommutative algebras, arXiv:1709.02720.

  84. [84]

    C. Curto and D.R. Morrison, Threefold flops via matrix factorization, J. Alg. Geom.22 (2013) 599.

    MathSciNet  MATH  Article  Google Scholar 

  85. [85]

    H.C. Pinkham, Factorization of birational maps in dimension 3, in Singularities, part 2 (Arcata, CA, U.S.A. 1981), Proc. Sympos. Pure Math.40, Amer. Math. Soc., Providence, RI, U.S.A. (1983), pg. 343.

  86. [86]

    M. Graña, R. Minasian, M. Petrini and A. Tomasiello, Generalized structures of N = 1 vacua, JHEP11 (2005) 020 [hep-th/0505212] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  87. [87]

    D. Xie and S.-T. Yau, Singularity, Sasaki-Einstein manifold, Log del Pezzo surface and N = 1 AdS/CFT correspondence: part I, arXiv:1903.00150 [INSPIRE].

  88. [88]

    D. Eisenbud and I. Peeva, Matrix factorizations for complete intersections and minimal free resolutions, arXiv:1306.2615.

  89. [89]

    D. Eisenbud and I. Peeva, Minimal free resolutions over complete intersections, Springer International Publishing, Cham, Switzerland (2016).

  90. [90]

    O. Iyama and M. Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math.197 (2013) 521.

    ADS  MathSciNet  MATH  Article  Google Scholar 

  91. [91]

    O. Iyama and M. Wemyss, Singular derived categories of Q-factorial terminalizations and maximal modification algebras, Adv. Math.261 (2014) 85.

    MathSciNet  MATH  Article  Google Scholar 

  92. [92]

    Á. Nolla de Celis and Y. Sekiya, Flops and mutations for crepant resolutions of polyhedral singularities, Asian J. Math.21 (2017) 1.

    MathSciNet  MATH  Article  Google Scholar 

  93. [93]

    W.M. Fairbairn, T. Fulton and W.H. Klink, Finite and disconnected subgroups of SU(3) and their application to the elementary-particle spectrum, J. Math. Phys.5 (1964) 1038 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

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Fazzi, M., Tomasiello, A. Holography, matrix factorizations and K-stability. J. High Energ. Phys. 2020, 119 (2020). https://doi.org/10.1007/JHEP05(2020)119

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Keywords

  • AdS-CFT Correspondence
  • Global Symmetries
  • Conformal Field Models in String Theory
  • Supersymmetric Gauge Theory