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Chiral four-dimensional F-theory compactifications with SU(5) and multiple U(1)-factors

  • Mirjam Cvetič
  • Antonella Grassi
  • Denis Klevers
  • Hernan Piragua
Open Access
Article

Abstract

We develop geometric techniques to determine the spectrum and the chiral indices of matter multiplets for four-dimensional F-theory compactifications on elliptic Calabi-Yau fourfolds with rank two Mordell-Weil group. The general elliptic fiber is the Calabi-Yau onefold in dP 2. We classify its resolved elliptic fibrations over a general base B. The study of singularities of these fibrations leads to explicit matter representations, that we determine both for U(1) × U(1) and SU(5) × U(1) × U(1) constructions. We determine for the first time certain matter curves and surfaces using techniques involving prime ideals. The vertical cohomology ring of these fourfolds is calculated for both cases and general formulas for the Euler numbers are derived. Explicit calculations are presented for a specific base B = ℙ3. We determine the general G 4-flux that belongs to \( H_V^{{\left( {2,2} \right)}} \) of the resolved Calabi-Yau fourfolds. As a by-product, we derive for the first time all conditions on G 4-flux in general F-theory compactifications with a non-holomorphic zero section. These conditions have to be formulated after a circle reduction in terms of Chern-Simons terms on the 3D Coulomb branch and invoke M-theory/F-theory duality. New Chern-Simons terms are generated by Kaluza-Klein states of the circle compactification. We explicitly perform the relevant field theory computations, that yield non-vanishing results precisely for fourfolds with a non-holomorphic zero section. Taking into account the new Chern-Simons terms, all 4D matter chiralities are determined via 3D M-theory/F-theory duality. We independently check these chiralities using the subset of matter surfaces we determined. The presented techniques are general and do not rely on toric data.

Keywords

Flux compactifications F-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]
    R. Donagi and M. Wijnholt, Model Building with F-theory, Adv. Theor. Math. Phys. 15 (2011) 1237 [arXiv:0802.2969] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory - I, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  3. [3]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory - II: Experimental Predictions, JHEP 01 (2009) 059 [arXiv:0806.0102] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    R. Donagi and M. Wijnholt, Breaking GUT Groups in F-theory, Adv. Theor. Math. Phys. 15 (2011) 1523 [arXiv:0808.2223] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    R. Blumenhagen, T.W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs, Nucl. Phys. B 829 (2010) 325 [arXiv:0908.1784] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    J. Marsano, N. Saulina and S. Schäfer-Nameki, Compact F-theory GUTs with U(1) (PQ), JHEP 04 (2010) 095 [arXiv:0912.0272] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    C.-M. Chen, J. Knapp, M. Kreuzer and C. Mayrhofer, Global SO(10) F-theory GUTs, JHEP 10 (2010) 057 [arXiv:1005.5735] [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    T.W. Grimm, S. Krause and T. Weigand, F-Theory GUT Vacua on Compact Calabi-Yau Fourfolds, JHEP 07 (2010) 037 [arXiv:0912.3524] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Knapp and M. Kreuzer, Toric Methods in F-theory Model Building, Adv. High Energy Phys. 2011 (2011) 513436 [arXiv:1103.3358] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  10. [10]
    J.J. Heckman, Particle Physics Implications of F-theory, Ann. Rev. Nucl. Part. Sci. 60 (2010) 237 [arXiv:1001.0577] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    A. Maharana and E. Palti, Models of Particle Physics from Type IIB String Theory and F-theory: A Review, Int. J. Mod. Phys. A 28 (2013) 1330005 [arXiv:1212.0555] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    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].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    K. Kodaira, On compact analytic surfaces: Ii, Annals Math. 77 (1963) 563.CrossRefzbMATHGoogle Scholar
  17. [17]
    J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular functions of one variable IV (1975) 33-52.Google Scholar
  18. [18]
    M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, arXiv:1107.0733 [INSPIRE].
  21. [21]
    J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and Spectral Covers from Resolved Calabi-Yaus, JHEP 11 (2011) 098 [arXiv:1108.1794] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    C. Lawrie and S. Schäfer-Nameki, The Tate Form on Steroids: Resolution and Higher Codimension Fibers, JHEP 04 (2013) 061 [arXiv:1212.2949] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    A. Grassi, J. Halverson and J.L. Shaneson, Matter From Geometry Without Resolution, JHEP 10 (2013) 205 [arXiv:1306.1832] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    S. Krause, C. Mayrhofer and T. Weigand, Gauge Fluxes in F-theory and Type IIB Orientifolds, JHEP 08 (2012) 119 [arXiv:1202.3138] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    R. Donagi and M. Wijnholt, Higgs Bundles and UV Completion in F-theory, Commun. Math. Phys. 326 (2014) 287 [arXiv:0904.1218] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. [26]
    J. Marsano, N. Saulina and S. Schäfer-Nameki, Monodromies, Fluxes and Compact Three-Generation F-theory GUTs, JHEP 08 (2009) 046 [arXiv:0906.4672] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    E. Dudas and E. Palti, Froggatt-Nielsen models from E 8 in F-theory GUTs, JHEP 01 (2010) 127 [arXiv:0912.0853] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  28. [28]
    M. Cvetič, I. Garcia-Etxebarria and J. Halverson, Global F-theory Models: Instantons and Gauge Dynamics, JHEP 01 (2011) 073 [arXiv:1003.5337] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    E. Dudas and E. Palti, On hypercharge flux and exotics in F-theory GUTs, JHEP 09 (2010) 013 [arXiv:1007.1297] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    M.J. Dolan, J. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory GUTs with U(1) Symmetries: Generalities and Survey, Phys. Rev. D 84 (2011) 066008 [arXiv:1102.0290] [INSPIRE].ADSGoogle Scholar
  31. [31]
    J. Marsano, H. Clemens, T. Pantev, S. Raby and H.-H. Tseng, A Global SU(5) F-theory model with Wilson line breaking, JHEP 01 (2013) 150 [arXiv:1206.6132] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  32. [32]
    A. Néron, Modeles minimaux des variétés abéliennes sur les corps locaux et globaux, Pub. Math. LIH ÉS 21 (1964) 5.CrossRefGoogle Scholar
  33. [33]
    T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Paul 39 (1990) 211.zbMATHMathSciNetGoogle Scholar
  34. [34]
    T. Shioda, Mordell-Weil lattices and Galois representation. I, Proc. Japan Acad. 65 A (1989) 268.CrossRefMathSciNetGoogle Scholar
  35. [35]
    R. Wazir, Arithmetic on Elliptic Threefolds, math.NT/0112259
  36. [36]
    J.H. Silverman, The arithmetic of elliptic curves, vol. 106, Springer (2009).Google Scholar
  37. [37]
    P.S. Aspinwall and D.R. Morrison, Nonsimply connected gauge groups and rational points on elliptic curves, JHEP 07 (1998) 012 [hep-th/9805206] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  38. [38]
    P.S. Aspinwall, S.H. Katz and D.R. Morrison, Lie groups, Calabi-Yau threefolds and F-theory, Adv. Theor. Math. Phys. 4 (2000) 95 [hep-th/0002012] [INSPIRE].zbMATHMathSciNetGoogle Scholar
  39. [39]
    T.W. Grimm and T. Weigand, On Abelian Gauge Symmetries and Proton Decay in Global F-theory GUTs, Phys. Rev. D 82 (2010) 086009 [arXiv:1006.0226] [INSPIRE].ADSGoogle Scholar
  40. [40]
    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].ADSCrossRefMathSciNetGoogle Scholar
  41. [41]
    S. Krause, C. Mayrhofer and T. Weigand, G 4 flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys. B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    T.W. Grimm and H. Hayashi, F-theory fluxes, Chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  43. [43]
    D.R. Morrison and D.S. Park, F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds, JHEP 10 (2012) 128 [arXiv:1208.2695] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  44. [44]
    M. Cvetič, T.W. Grimm and D. Klevers, Anomaly Cancellation And Abelian Gauge Symmetries In F-theory, JHEP 02 (2013) 101 [arXiv:1210.6034] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    V. Braun, T.W. Grimm and J. Keitel, New Global F-theory GUTs with U(1) symmetries, JHEP 09 (2013) 154 [arXiv:1302.1854] [INSPIRE].ADSMathSciNetGoogle Scholar
  46. [46]
    J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, Elliptic fibrations for SU(5) × U(1) × U(1) F-theory vacua, Phys. Rev. D 88 (2013) 046005 [arXiv:1303.5054] [INSPIRE].ADSGoogle Scholar
  47. [47]
    V. Braun, T.W. Grimm and J. Keitel, Geometric Engineering in Toric F-theory and GUTs with U(1) Gauge Factors, JHEP 12 (2013) 069 [arXiv:1306.0577] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    A. Klemm, M. Kreuzer, E. Riegler and E. Scheidegger, Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections, JHEP 05 (2005) 023 [hep-th/0410018] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  49. [49]
    M. Cvetič, D. Klevers, H. Piragua and P. Song, Elliptic Fibrations with Rank Three Mordell-Weil Group: F-theory with U(1) × U(1) x U(1) Gauge Symmetry, arXiv:1310.0463 [INSPIRE].
  50. [50]
    A. Grassi and V. Perduca, Weierstrass models of elliptic toric K3 hypersurfaces and symplectic cuts, arXiv:1201.0930 [INSPIRE].
  51. [51]
    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
  52. [52]
    G. Aldazabal, A. Font, L.E. Ibáñez and A.M. Uranga, New branches of string compactifications and their F-theory duals, Nucl. Phys. B 492 (1997) 119 [hep-th/9607121] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, hep-th/9607139 [INSPIRE].
  54. [54]
    M. Cvetič, D. Klevers and H. Piragua, F-Theory Compactifications with Multiple U(1)-Factors: Addendum, JHEP 12 (2013) 056 [arXiv:1307.6425] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    T.W. Grimm, A. Kapfer and J. Keitel, Effective action of 6D F-theory with U(1) factors: Rational sections make Chern-Simons terms jump, JHEP 07 (2013) 115 [arXiv:1305.1929] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  56. [56]
    P. Mayr, Mirror symmetry, N = 1 superpotentials and tensionless strings on Calabi-Yau four folds, Nucl. Phys. B 494 (1997) 489 [hep-th/9610162] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  57. [57]
    A. Klemm, B. Lian, S.S. Roan and S.-T. Yau, Calabi-Yau fourfolds for M-theory and F-theory compactifications, Nucl. Phys. B 518 (1998) 515 [hep-th/9701023] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  58. [58]
    T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Computing Brane and Flux Superpotentials in F-theory Compactifications, JHEP 04 (2010) 015 [arXiv:0909.2025] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  59. [59]
    D.S. Park, Anomaly Equations and Intersection Theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    T.W. Grimm and R. Savelli, Gravitational Instantons and Fluxes from M/F-theory on Calabi-Yau fourfolds, Phys. Rev. D 85 (2012) 026003 [arXiv:1109.3191] [INSPIRE].ADSGoogle Scholar
  61. [61]
    F. Bonetti and T.W. Grimm, Six-dimensional (1,0) effective action of F-theory via M-theory on Calabi-Yau threefolds, JHEP 05 (2012) 019 [arXiv:1112.1082] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  62. [62]
    D. Klevers, Holomorphic Couplings In Non-Perturbative String Compactifications, Fortsch. Phys. 60 (2012) 3 [arXiv:1106.6259] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  63. [63]
    B.R. Greene, D.R. Morrison and M.R. Plesser, Mirror manifolds in higher dimension, Commun. Math. Phys. 173 (1995) 559 [hep-th/9402119] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  64. [64]
    S. Sethi, C. Vafa and E. Witten, Constraints on low dimensional string compactifications, Nucl. Phys. B 480 (1996) 213 [hep-th/9606122] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  65. [65]
    K. Dasgupta, G. Rajesh and S. Sethi, M theory, orientifolds and G - flux, JHEP 08 (1999) 023 [hep-th/9908088] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  66. [66]
    C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Contact Terms, Unitarity and F-Maximization in Three-Dimensional Superconformal Theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  67. [67]
    C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons Contact Terms in Three Dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  68. [68]
    F. Bonetti, T.W. Grimm and S. Hohenegger, A Kaluza-Klein inspired action for chiral p-forms and their anomalies, Phys. Lett. B 720 (2013) 424 [arXiv:1206.1600] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  69. [69]
    E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  70. [70]
    S. Gukov, C. Vafa and E. Witten, CFTs from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477-478] [hep-th/9906070] [INSPIRE].
  71. [71]
    A. Bilal and S. Metzger, Anomaly cancellation in M-theory: A Critical review, Nucl. Phys. B 675 (2003) 416 [hep-th/0307152] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  72. [72]
    K. Intriligator, H. Jockers, P. Mayr, D.R. Morrison and M.R. Plesser, Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes, arXiv:1203.6662 [INSPIRE].
  73. [73]
    T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys. B 845 (2011) 48 [arXiv:1008.4133] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  74. [74]
    T.W. Grimm, D. Klevers and M. Poretschkin, Fluxes and Warping for Gauge Couplings in F-theory, JHEP 01 (2013) 023 [arXiv:1202.0285] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  75. [75]
    A.J. Niemi and G.W. Semenoff, Axial Anomaly Induced Fermion Fractionization and Effective Gauge Theory Actions in Odd Dimensional Space-Times, Phys. Rev. Lett. 51 (1983) 2077 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  76. [76]
    A.N. Redlich, Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].ADSMathSciNetGoogle Scholar
  77. [77]
    O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  78. [78]
    H. Hayashi, C. Lawrie and S. Schäfer-Nameki, Phases, Flops and F-theory: SU(5) Gauge Theories, JHEP 10 (2013) 046 [arXiv:1304.1678] [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  80. [80]
    H. Hayashi, R. Tatar, Y. Toda, T. Watari and M. Yamazaki, New Aspects of Heterotic-F Theory Duality, Nucl. Phys. B 806 (2009) 224 [arXiv:0805.1057] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  81. [81]
    M.B. Green and J.H. Schwarz, Anomaly Cancellation in Supersymmetric D = 10 Gauge Theory and Superstring Theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  82. [82]
    A. Sagnotti, A Note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B 294 (1992) 196 [hep-th/9210127] [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    T.W. Grimm and W. Taylor, Structure in 6D and 4D N = 1 supergravity theories from F-theory, JHEP 10 (2012) 105 [arXiv:1204.3092] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  84. [84]
    P. Candelas, D.-E. Diaconescu, B. Florea, D.R. Morrison and G. Rajesh, Codimension three bundle singularities in F-theory, JHEP 06 (2002) 014 [hep-th/0009228] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  85. [85]
    M. Cvetic, A. Grassi, D. Klevers, and H. Piragua, work in progress.Google Scholar
  86. [86]
    D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  87. [87]
    D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Mirjam Cvetič
    • 1
    • 2
  • Antonella Grassi
    • 3
  • Denis Klevers
    • 1
  • Hernan Piragua
    • 1
  1. 1.Department of Physics and AstronomyUniversity of PennsylvaniaPhiladelphiaU.S.A.
  2. 2.Center for Applied Mathematics and Theoretical PhysicsUniversity of MariborMariborSlovenia
  3. 3.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaU.S.A.

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