Journal of High Energy Physics

, 2013:112 | Cite as

Refined stable pair invariants for E-, M- and [p, q]-strings

  • Min-xin HuangEmail author
  • Albrecht Klemm
  • Maximilian Poretschkin


We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the fivedimensional N =1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N = 2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p, q]-strings as well as M-strings.


Differential and Algebraic Geometry Topological Strings M-Theory 


  1. [1]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  2. [2]
    J. Choi, S. Katz and A. Klemm, The refined BPS index from stable pair invariants, arXiv:1210.4403 [INSPIRE].
  3. [3]
    N. Nekrasov and A. Okounkov, The M-theory index, in preparation.Google Scholar
  4. [4]
    J. Minahan, D. Nemeschansky and N. Warner, Partition functions for BPS states of the noncritical E 8 string, Adv. Theor. Math. Phys. 1 (1998) 167 [hep-th/9707149] [INSPIRE].MathSciNetGoogle Scholar
  5. [5]
    J. Minahan, D. Nemeschansky and N. Warner, Investigating the BPS spectrum of noncritical E(n) strings, Nucl. Phys. B 508 (1997) 64 [hep-th/9705237] [INSPIRE].MathSciNetADSGoogle Scholar
  6. [6]
    A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, hep-th/9607139 [INSPIRE].
  7. [7]
    J. Minahan, D. Nemeschansky, C. Vafa and N. Warner, E strings and N = 4 topological Yang-Mills theories, Nucl. Phys. B 527 (1998) 581 [hep-th/9802168] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    T. Eguchi and K. Sakai, Seiberg-Witten curve for E string theory revisited, Adv. Theor. Math. Phys. 7 (2004) 419 [hep-th/0211213] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  9. [9]
    K. Hori et al., Clay Mathmatics Monographs. Vol. 1: Mirror symmetry, AMS Publications, Rhode Island U.S.A. (2003).Google Scholar
  10. [10]
    E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    O.J. Ganor, A test of the chiral E 8 current algebra on a 6 − D noncritical string, Nucl. Phys. B 479 (1996) 197 [hep-th/9607020] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, Del Pezzo surfaces and type-I-prime theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    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].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, arXiv:1306.1734 [INSPIRE].
  19. [19]
    M. Demazure, Surfaces de del Pezzo - I-V, in Lecture Notes in Mathematics: Vol. 777: Séminaire sur les Singularités des Surfaces, Springer, Berlin Germany (1980).Google Scholar
  20. [20]
    I. Dolgachev, Classical algebraic geometry: a modern view, Cambridge University Press, Cambridge U.K. (2012).CrossRefGoogle Scholar
  21. [21]
    K. Saito, Einfach-elliptische Singularitäten, Invent. Math. 23 (1974) 289.MathSciNetADSCrossRefzbMATHGoogle Scholar
  22. [22]
    E. Looijenga, Root Systems and Elliptic Curves, Invent. Math. 38 (1977) 17.MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    E. Looijenga, On the semi-universal deformation of a simple elliptic hypersurface singularity, Topology 17 (1978) 23.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    E. Looijenga, Invariant Theory for generalized Root Systems, Invent. Math. 61 (1980) 1.MathSciNetADSCrossRefzbMATHGoogle Scholar
  25. [25]
    K. Saito, Extended affine root systems II, Publ. Rims. Kyoto Univ. 26 (1990) 15.CrossRefzbMATHGoogle Scholar
  26. [26]
    K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo 27 (1980) 265.zbMATHGoogle Scholar
  27. [27]
    I.N. Bernshtein and O.V. Shvartsman, Chevalleys theorem for complex crystallographic Coxeter Groups, Funct. Anal. Appl. 12 (1978) 308.CrossRefGoogle Scholar
  28. [28]
    K. Wirthmüller, Root systems and Jacobi forms, Compos. Math. 82 (1992) 293.zbMATHGoogle Scholar
  29. [29]
    R. Pandharipande and R. Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009) 407 [arXiv:0707.2348] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  30. [30]
    R. Pandharipande and R. Thomas, The 3-fold vertex via stable pairs, arXiv:0709.3823 [INSPIRE].
  31. [31]
    R. Pandharipande and R. Thomas, Stable pairs and BPS invariants, arXiv:0711.3899 [INSPIRE].
  32. [32]
    R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  33. [33]
    A. Johansen, A comment on BPS states in F-theory in eight-dimensions, Phys. Lett. B 395 (1997) 36 [hep-th/9608186] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    M.R. Gaberdiel and B. Zwiebach, Exceptional groups from open strings, Nucl. Phys. B 518 (1998) 151 [hep-th/9709013] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    M.R. Gaberdiel, T. Hauer and B. Zwiebach, Open string-string junction transitions, Nucl. Phys. B 525 (1998) 117 [hep-th/9801205] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    O. DeWolfe, T. Hauer, A. Iqbal and B. Zwiebach, Uncovering the symmetries on [p,q] seven-branes: Beyond the Kodaira classification, Adv. Theor. Math. Phys. 3 (1999) 1785 [hep-th/9812028] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  38. [38]
    B. Haghighat, A. Iqbal, C. Kozcaz, G. Lockhart and C. Vafa, M-Strings, arXiv:1305.6322 [INSPIRE].
  39. [39]
    P.C. Argyres, M.R. Plesser and A.D. Shapere, The Coulomb phase of N = 2 supersymmetric QCD, Phys. Rev. Lett. 75 (1995) 1699 [hep-th/9505100] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  40. [40]
    M. Aganagic, M.C. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum geometry of refined topological strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The Topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  42. [42]
    M. Aganagic and K. Schaeffer, Refined black hole ensembles and topological strings, JHEP 01 (2013) 060 [arXiv:1210.1865] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    M. Aganagic and S. Shakirov, Refined Chern-Simons Theory and Topological String, arXiv:1210.2733 [INSPIRE].
  44. [44]
    M. Alim and E. Scheidegger, Topological Strings on Elliptic Fibrations, arXiv:1205.1784 [INSPIRE].
  45. [45]
    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2., Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    H. Awata and H. Kanno, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  47. [47]
    V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  48. [48]
    K. Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. Math. 170 (2009) 1307 math/0507523.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  50. [50]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    A. Strominger, Open p-branes, Phys. Lett. B 383 (1996) 44 [hep-th/9512059] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    J. Fuchs, Affine Lie Algebras and Quantum Groups, Cambridge University Press, Cambridge U.K. (1992).zbMATHGoogle Scholar
  54. [54]
    R. Donagi, Tanaguchi Lectures on Principal Bundles on Elliptic Fibrations, hep-th/98020994.
  55. [55]
    A. Klemm, W. Lerche and S. Theisen, Nonperturbative effective actions of N = 2 supersymmetric gauge theories, Int. J. Mod. Phys. A 11 (1996) 1929 [hep-th/9505150] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral Four-Dimensional F-theory Compactifications With SU(5) and Multiple U(1)-Factors, arXiv:1306.3987 [INSPIRE].
  57. [57]
    M. Cvetič, D. Klevers and H. Piragua, F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections, JHEP 06 (2013) 067 [arXiv:1303.6970] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral Four-Dimensional F-theory Compactifications With SU(5) and Multiple U(1)-Factors, arXiv:1306.3987 [INSPIRE].
  59. [59]
    P. Hořava and E. Witten, Eleven-dimensional supergravity on a manifold with boundary, Nucl. Phys. B 475 (1996) 94 [hep-th/9603142] [INSPIRE].ADSGoogle Scholar
  60. [60]
    P. Hořava and E. Witten, Heterotic and type-I string dynamics from eleven-dimensions, Nucl. Phys. B 460 (1996) 506 [hep-th/9510209] [INSPIRE].ADSGoogle Scholar
  61. [61]
    M. Vasiliev, Higher spin superalgebras in any dimension and their representations, JHEP 12 (2004) 046 [hep-th/0404124] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    M. Vasiliev, Higher spin gauge theories in various dimensions, Fortsch. Phys. 52 (2004) 702 [hep-th/0401177] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  63. [63]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, Class. Quant. Grav. 30 (2013) 104003 [arXiv:1204.3882] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  64. [64]
    J. Maldacena and A. Zhiboedov, Constraining Conformal Field Theories with A Higher Spin Symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].MathSciNetADSGoogle Scholar
  65. [65]
    D.A. Cox, The Homogeneous coordinate ring of a toric variety, revised version, alg-geom/9210008 [INSPIRE].
  66. [66]
    R. Dijkgraaf, Mirror symmetry and elliptic curves, in The Moduli Space of Curves, Progr. Math. 129 (1995) 149.MathSciNetGoogle Scholar
  67. [67]
    T. Eguchi and K. Sakai, Seiberg-Witten curve for the E string theory, JHEP 05 (2002) 058 [hep-th/0203025] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  68. [68]
    T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, The D5-brane effective action and superpotential in N = 1 compactifications, Nucl. Phys. B 816 (2009) 139 [arXiv:0811.2996] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  69. [69]
    A. Néron, Propriètés arithmétiques de certaines familles de courbes algébriques, in Proc. Int. Congress, Amsterdam The Netherlands (1954), pg. 481.Google Scholar
  70. [70]
    A. Néron, Les propriètés du rang des courbes algébriques dans les corps de degré de transcendance fini, Colloques Internationaux du Centre National de la Recherche Scientifique. Vol. 24, Paris France (1950), pg. 65.Google Scholar
  71. [71]
    C.F. Schwartz, An elliptic surface of Mordell-Weil rank 8 over the rational numbers, J. Theor. Nombres Bordeaux 6 (1994) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    J.I. Manin, The Tate height of points on an Abelian variety, its variants and applications, AMS Trans. 59 (1966) 82.Google Scholar
  73. [73]
    W. Fulton, Annals of Math. Studies. Vol. 131: Introduction to toric varieties, Princeton University Press, Princeton U.S.A. (1993).Google Scholar
  74. [74]
    R. Gopakumar and C. Vafa, M theory and topological strings. 1., hep-th/9809187 [INSPIRE].
  75. [75]
    R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].
  76. [76]
    L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990) 193.MathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    T. Chiang, A. Klemm, S.-T. Yau and E. Zaslow, Local mirror symmetry: Calculations and interpretations, Adv. Theor. Math. Phys. 3 (1999) 495 [hep-th/9903053] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  78. [78]
    A. Klemm and E. Zaslow, Local mirror symmetry at higher genus, hep-th/9906046 [INSPIRE].
  79. [79]
    S. Hosono, Counting BPS states via holomorphic anomaly equations, Fields Inst. Commun. (2002) 57 [hep-th/0206206] [INSPIRE].
  80. [80]
    S. Hosono, M. Saito and A. Takahashi, Holomorphic anomaly equation and BPS state counting of rational elliptic surface, Adv. Theor. Math. Phys. 3 (1999) 177 [hep-th/9901151] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  81. [81]
    M.-x. Huang, A.-K. Kashani-Poor and A. Klemm, The Ω deformed B-model for rigid \( \mathcal{N} \) = 2 theories, Annales Henri Poincaré 14 (2013) 425 [arXiv:1109.5728] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  82. [82]
    M.-x. Huang and A. Klemm, Holomorphic Anomaly in Gauge Theories and Matrix Models, JHEP 09 (2007) 054 [hep-th/0605195] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  83. [83]
    M.-x. Huang and A. Klemm, Direct integration for general Ω backgrounds, Adv. Theor. Math. Phys. 16 (2012), no. 3 805–849 [arXiv:1009.1126] [INSPIRE].
  84. [84]
    M.-x. Huang, A. Klemm and S. Quackenbush, Topological string theory on compact Calabi-Yau: Modularity and boundary conditions, Lect. Notes Phys. 757 (2009) 45 [hep-th/0612125] [INSPIRE].MathSciNetADSGoogle Scholar
  85. [85]
    A. Iqbal and C. Kozcaz, Refined Topological Strings and Toric Calabi-Yau Threefolds, arXiv:1210.3016 [INSPIRE].
  86. [86]
    A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  87. [87]
    M. Kaneko and D.B. Zagier, A generalized Jacobi theta function and quasi-modular forms, in The Moduli Space of Curves, Progr. Math. 129 (1995) 165.MathSciNetGoogle Scholar
  88. [88]
    S.H. Katz, A. Klemm and C. Vafa, M theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999) 1445 [hep-th/9910181] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  89. [89]
    A. Klemm, J. Manschot and T. Wotschke, Quantum geometry of elliptic Calabi-Yau manifolds, arXiv:1205.1795 [INSPIRE].
  90. [90]
    B. Haghighat, A. Klemm and M. Rauch, Integrability of the holomorphic anomaly equations, JHEP 10 (2008) 097 [arXiv:0809.1674] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  91. [91]
    A. Iqbal, C. Kozcaz and K. Shabbir, Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory, Nucl. Phys. B 838 (2010) 422 [arXiv:0803.2260] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  92. [92]
    D. Krefl and J. Walcher, Extended Holomorphic Anomaly in Gauge Theory, Lett. Math. Phys. 95 (2011) 67 [arXiv:1007.0263] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  93. [93]
    D. Krefl and J. Walcher, Shift versus Extension in Refined Partition Functions, arXiv:1010.2635 [INSPIRE].
  94. [94]
    F. Klein, Vorlesungen über die Theorie der elliptischen Modulfunktionen, Teubner, Leipzig Germany (1890).Google Scholar
  95. [95]
    R. Pandharipande and R. Thomas, The 3-fold vertex via stable pairs, arXiv:0709.3823 [INSPIRE].
  96. [96]
    T. Oda, Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties, Springer, Berlin Germany (1988).zbMATHGoogle Scholar
  97. [97]
    K. Sakai, Topological string amplitudes for the local half K3 surface, arXiv:1111.3967 [INSPIRE].
  98. [98]
    C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  99. [99]
    K. Yoshioka, Euler characteristics of SU(2) instanton moduli spaces on rational elliptic surfaces, Commun. Math. Phys. 205 (1999) 501 [math/9805003] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  100. [100]
    I. Connell, Elliptic Curve Handbook,
  101. [101]
    J.J. Duistermaat, Discrete Integrable Systems, Springer Monographs in Mathematics, Springer, Heidelberg Germany (2010).Google Scholar
  102. [102]
    R. Donagi, S. Katz and M. Wijnholt, Weak Coupling, Degeneration and Log Calabi-Yau Spaces, arXiv:1212.0553 [INSPIRE].
  103. [103]
    P. Slowody, Lecture Notes in Mathematics. Vol. 815: Simple singularities and simple algebraic groups, Springer, Heidelberg Germany (1980).Google Scholar
  104. [104]
    C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  105. [105]
    M. Alim et al., Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes, arXiv:1012.1608 [INSPIRE].
  106. [106]
    V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287 (2009) 117 [arXiv:0709.1453] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  107. [107]
    M. Mariño, Open string amplitudes and large order behavior in topological string theory, JHEP 03 (2008) 060 [hep-th/0612127] [INSPIRE].ADSCrossRefGoogle Scholar
  108. [108]
    A. Klemm, M. Mariño, M. Schiereck and M. Soroush, ABJM Wilson loops in the Fermi gas approach, arXiv:1207.0611 [INSPIRE].
  109. [109]
    W. Lerche and S. Stieberger, Prepotential, mirror map and F-theory on K3, Adv. Theor. Math. Phys. 2 (1998) 1105 [Erratum ibid. 3 (1999) 1199] [hep-th/9804176] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  110. [110]
    W. Lerche, S. Stieberger and N. Warner, Quartic gauge couplings from K3 geometry, Adv. Theor. Math. Phys. 3 (1999) 1575 [hep-th/9811228] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  111. [111]
    K. Dasgupta and S. Mukhi, BPS nature of three string junctions, Phys. Lett. B 423 (1998) 261 [hep-th/9711094] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  112. [112]
    A. Mikhailov, N. Nekrasov and S. Sethi, Geometric realizations of BPS states in N = 2 theories, Nucl. Phys. B 531 (1998) 345 [hep-th/9803142] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  113. [113]
    A. Klemm, W. Lerche, P. Mayr, C. Vafa and N.P. Warner, Selfdual strings and N = 2 supersymmetric field theory, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  114. [114]
    K. Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge University Press, Cambridge U.K. (2002).CrossRefzbMATHGoogle Scholar
  115. [115]
    S.-T. Yau and E. Zaslow, BPS states, string duality and nodal curves on K3, Nucl. Phys. B 471 (1996) 503 [hep-th/9512121] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • Min-xin Huang
    • 1
    Email author
  • Albrecht Klemm
    • 2
    • 3
  • Maximilian Poretschkin
    • 2
  1. 1.Interdisciplinary Center for Theoretical StudyUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Bethe Center for Theoretical PhysicsUniversität BonnBonnGermany
  3. 3.Hausdorff Center for MathematicsUniversität BonnBonnGermany

Personalised recommendations