Advertisement

Dai-Freed anomalies in particle physics

  • Iñaki García-EtxebarriaEmail author
  • Miguel Montero
Open Access
Regular Article - Theoretical Physics

Abstract

Anomalies can be elegantly analyzed by means of the Dai-Freed theorem. In this framework it is natural to consider a refinement of traditional anomaly cancellation conditions, which sometimes leads to nontrivial extra constraints in the fermion spectrum. We analyze these more refined anomaly cancellation conditions in a variety of theories of physical interest, including the Standard Model and the SU(5) and Spin(10) GUTs, which we find to be anomaly free. Turning to discrete symmetries, we find that baryon triality has a ℤ9 anomaly that only cancels if the number of generations is a multiple of 3. Assuming the existence of certain anomaly-free ℤ4 symmetry we relate the fact that there are 16 fermions per generation of the Standard model — including right-handed neutrinos — to anomalies under time-reversal of boundary states in four-dimensional topological superconductors. A similar relation exists for the MSSM, only this time involving the number of gauginos and Higgsinos, and it is non-trivially, and remarkably, satisfied for the SU(3) × SU(2) × U(1) gauge group with two Higgs doublets. We relate the constraints we find to the well-known Ibañez-Ross ones, and discuss the dependence on UV data of the construction. Finally, we comment on the (non-)existence of K-theoretic θ angles in four dimensions.

Keywords

Anomalies in Field and String Theories Discrete Symmetries Gauge Symmetry Differential and Algebraic Geometry 

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]
    G. ’t Hooft et al., Recent developments in gauge theories. Proceedings, Nato Advanced Study Institute, Cargese, France, August 26 - September 8, 1979, NATO Sci. Ser.B 59 (1980) 1.Google Scholar
  2. [2]
    E. Witten, An SU(2) anomaly, Phys. Lett.B 117 (1982) 324.ADSMathSciNetGoogle Scholar
  3. [3]
    A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, JHEP12 (2015) 052 [arXiv:1406.7329] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  4. [4]
    C.-T. Hsieh, G.Y. Cho and S. Ryu, Global anomalies on the surface of fermionic symmetry-protected topological phases in (3 + 1) dimensions, Phys. Rev.B 93 (2016) 075135 [arXiv:1503.01411] [INSPIRE].ADSGoogle Scholar
  5. [5]
    E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys.88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].ADSGoogle Scholar
  6. [6]
    D.S. Freed, Pions and generalized cohomology, J. Diff. Geom.80 (2008) 45 [hep-th/0607134] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  7. [7]
    C.-T. Hsieh, Discrete gauge anomalies revisited, arXiv:1808.02881 [INSPIRE].
  8. [8]
    E. Witten, Global gravitational anomalies, Commun. Math. Phys.100 (1985) 197 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  9. [9]
    L. Álvarez-Gaumé, S. Della Pietra and G.W. Moore, Anomalies and odd dimensions, Annals Phys.163 (1985) 288 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  10. [10]
    A. Bilal, Lectures on anomalies, arXiv:0802.0634 [INSPIRE].
  11. [11]
    A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge U.K. (2000).zbMATHGoogle Scholar
  12. [12]
    D.S. Freed, Determinants, torsion and strings, Commun. Math. Phys.107 (1986) 483 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  13. [13]
    I. García-Etxebarria et al., 8d gauge anomalies and the topological Green-Schwarz mechanism, JHEP11 (2017) 177 [arXiv:1710.04218] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  14. [14]
    L. Álvarez-Gaumé and P.H. Ginsparg, The topological meaning of nonabelian anomalies, Nucl. Phys.B 243 (1984) 449 [INSPIRE].ADSMathSciNetGoogle Scholar
  15. [15]
    K. Yonekura, Dai-Freed theorem and topological phases of matter, JHEP09 (2016) 022 [arXiv:1607.01873] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  16. [16]
    M.F. Atiyah and I.M. Singer, The index of elliptic operators. 1, Annals Math.87 (1968) 484.MathSciNetzbMATHGoogle Scholar
  17. [17]
    X.-z. Dai and D.S. Freed, η invariants and determinant lines, J. Math. Phys.35 (1994) 5155 [Erratum ibid.42 (2001) 2343] [hep-th/9405012] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  18. [18]
    M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Phil. Soc.77 (1975) 43.ADSMathSciNetzbMATHGoogle Scholar
  19. [19]
    M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry II, Math. Proc. Cambridge Phil. Soc.78 (1976) 405.MathSciNetzbMATHGoogle Scholar
  20. [20]
    M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry III, Math. Proc. Cambridge Phil. Soc.79 (1976) 71.ADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    S. Monnier, The global anomalies of (2, 0) superconformal field theories in six dimensions, JHEP09 (2014) 088 [arXiv:1406.4540] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  22. [22]
    S.B. Giddings and A. Strominger, Axion induced topology change in quantum gravity and string theory, Nucl. Phys.B 306 (1988) 890 [INSPIRE].ADSGoogle Scholar
  23. [23]
    E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys.2 (1998) 505 [hep-th/9803131] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  24. [24]
    D.S. Freed, Anomalies and invertible field theories, Proc. Symp. Pure Math.88 (2014) 25 [arXiv:1404.7224] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  25. [25]
    S. Monnier, A modern point of view on anomalies, arXiv:1903.02828 [INSPIRE].
  26. [26]
    F. Benini, C. Córdova and P.-S. Hsin, On 2-group global symmetries and their anomalies, JHEP03 (2019) 118 [arXiv:1803.09336] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  27. [27]
    L. Nicolaescu, Notes on Seiberg-Witten theory, Graduate studies in mathematics, American Mathematical Society, U.S.A. (2000).Google Scholar
  28. [28]
    M. Guo, P. Putrov and J. Wang, Time reversal, SU(N) Yang-Mills and cobordisms: interacting topological superconductors/insulators and quantum spin liquids in 3 + 1D, Annals Phys.394 (2018) 244 [arXiv:1711.11587] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  29. [29]
    J. Wang et al., Tunneling topological vacua via extended operators: (spin-)TQFT spectra and boundary deconfinement in various dimensions, PTEP2018 (2018) 053A01 [arXiv:1801.05416] [INSPIRE].Google Scholar
  30. [30]
    J. McCleary, A users guide to spectral sequences, 2nd edition, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge U.K. (2000).Google Scholar
  31. [31]
    J. Davis and P. Kirk, Lecture notes in algebraic topology, Graduate studies in mathematics, American Mathematical Society, U.S.A. (2001).zbMATHGoogle Scholar
  32. [32]
    D.W. Anderson, E.H. Brown and F.P. Peterson, Spin cobordism, Bull. Amer. Math. Soc.72 (1966) 256.MathSciNetzbMATHGoogle Scholar
  33. [33]
    R. Stong, Calculation of \( {\varOmega}_{11}^{\mathrm{spin}} \) (K(Z, 4)), in Unified string theories, M. Green and D. Gross eds., World Scientific, Singapore (1986).Google Scholar
  34. [34]
    P. Teichner, Topological four-manifolds with finite fundamental group, Ph.D. thesis, Johannes-Gutenberg Universität, Mainz, Germany (1992).Google Scholar
  35. [35]
    P. Teichner, On the signature of four-manifolds with universal covering spin, Math. Ann. (1993) 745.Google Scholar
  36. [36]
    J. Adams and J. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, U.S.A. (1995).Google Scholar
  37. [37]
    W.T. Wu, Les I-carrés dans une variété grassmannienne, World Scientific, Singapore (2012).zbMATHGoogle Scholar
  38. [38]
    A. Borel, La cohomologie mod 2 de certains espaces homogènes, Comm. Math. Helv.27 (1953) 165.zbMATHGoogle Scholar
  39. [39]
    E. Milnor et al., Characteristic Classes, Annals of Mathematics Studies, Princeton University Press, Princeton U.S.A. (1974).Google Scholar
  40. [40]
    J.H. Fung, The Cohomology of Lie groups, http://math.uchicago.edu/~may/REU2012/REUPapers/Fung.pdf.
  41. [41]
    K. Marathe, Topics in physical mathematics: geometric topology and field theory, Springer, Germany (2010).zbMATHGoogle Scholar
  42. [42]
    R. Switzer, Algebraic topology: homotopy and homology, Classics in Mathematics, Springer, Germany (2002).zbMATHGoogle Scholar
  43. [43]
    K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys.B 497 (1997) 56 [hep-th/9702198] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  44. [44]
    A.N. Redlich, Gauge noninvariance and parity violation of three-dimensional fermions, Phys. Rev. Lett.52 (1984) 18 [INSPIRE].ADSMathSciNetGoogle Scholar
  45. [45]
    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
  46. [46]
    D. Husemöller, M. Joachim, B. Jurco and M. Schottenloher, Basic bundle theory and K-cohomology invariants, Lecture Notes in Physics, Springer, Germany (2007).zbMATHGoogle Scholar
  47. [47]
    D.-E. Diaconescu, G.W. Moore and E. Witten, E 8gauge theory and a derivation of k-theory from M-theory, Adv. Theor. Math. Phys.6 (2003) 1031 [hep-th/0005090] [INSPIRE].Google Scholar
  48. [48]
    A. Bahri and P. Gilkey, The eta invariant, pincbordism, and equivariant spincbordism for cyclic 2-groups., Pacific J. Math.128 (1987) 1.MathSciNetzbMATHGoogle Scholar
  49. [49]
    E. Witten, Topological tools in ten-dimensional physics, Int. J. Mod. Phys.A 1 (1986) 39 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  50. [50]
    D. Fuchs and O. Viro, Topology II: homotopy and homology. classical manifolds, Encyclopaedia of Mathematical Sciences, Springer, Germany (2003).Google Scholar
  51. [51]
    A. Borel, Topology of lie groups and characteristic classes, Bull. Amer. Math. Soc.61 (1955) 397.MathSciNetzbMATHGoogle Scholar
  52. [52]
    A. Borel and J.P. Serre, Groupes de Lie et puissances reduites de Steenrod, Amer. J. Math.75 (1953) 409.MathSciNetzbMATHGoogle Scholar
  53. [53]
    D. Tong, Line operators in the standard model, JHEP07 (2017) 104 [arXiv:1705.01853] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  54. [54]
    S. Monnier, G.W. Moore and D.S. Park, Quantization of anomaly coefficients in 6D \( \mathcal{N} \) = (1,0) supergravity, JHEP02(2018) 020 [arXiv:1711.04777] [INSPIRE].ADSGoogle Scholar
  55. [55]
    K.-w. Choi, D.B. Kaplan and A.E. Nelson, Is CP a gauge symmetry?, Nucl. Phys.B 391 (1993) 515 [hep-ph/9205202] [INSPIRE].
  56. [56]
    P. Gilkey, The geometry of spherical space form groups, Series in pure mathematics, World Scientific, Singapore (1989).Google Scholar
  57. [57]
    M. Berg, C. DeWitt-Morette, S. Gwo and E. Kramer, The pin groups in physics: C, P and T, Rev. Math. Phys.13 (2001) 953 [math-ph/0012006] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  58. [58]
    Y. Tachikawa and K. Yonekura, Why are fractional charges of orientifolds compatible with Dirac quantization?, arXiv:1805.02772 [INSPIRE].
  59. [59]
    X. Gu, On the cohomology of the classifying spaces of projective unitary groups, arXiv:1612.00506.
  60. [60]
    J. Rotman, An introduction to algebraic topology, Graduate Texts in Mathematics, Springer, Germany (1998).Google Scholar
  61. [61]
    L. Breen, R. Mikhailov and A. Touzé, Derived functors of the divided power functors, arXiv:1312.5676.
  62. [62]
    H. Cartan, Séminaire Henri Cartan de lEcole Normale Supérieure, 1954/1955. Algèbres dEilenberg-MacLane et homotopie, (1967).Google Scholar
  63. [63]
    N. Pointet-Tischler, La suspension cohomologique des espaces dEilenberg-MacLane, Compt. Rend. Acad. Sci. Ser.I 325 (1997) 1113.MathSciNetzbMATHGoogle Scholar
  64. [64]
    N. Tischler, Invariants de Postnikov des espaces de lacets, Ph.D. thesis, Université de Lausanne, Lausanne, Switzerland (1996).Google Scholar
  65. [65]
    A. Clément, Integral cohomology of finite Postnikov towers, Ph.D. thesis, Université de Lausanne, Lausanne, Switzerland (2002).Google Scholar
  66. [66]
    M. Feshbach, The integral cohomology rings of the classifying spaces of o(n) and so(n), Indiana Univ. Math. J.32 (1983) 511.MathSciNetzbMATHGoogle Scholar
  67. [67]
    E. H. Brown, The cohomology of BSO nand BO nwith integer coefficients, Proc. Amer. Math. Soc.85 (1982) 283.MathSciNetGoogle Scholar
  68. [68]
    M. Feshbach, The integral cohomology rings of the classifying spaces of o(n) and so(n), Indiana Univ, Math. Lett.32 (1983) 511.MathSciNetzbMATHGoogle Scholar
  69. [69]
    D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups., Math. Ann.194 (1971) 197.MathSciNetzbMATHGoogle Scholar
  70. [70]
    A. Kono, On the integral cohomology of bspin(n), J. Math. Kyoto Univ.26 (1986) 333.MathSciNetzbMATHGoogle Scholar
  71. [71]
    M. Kameko and M. Mimura, On the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of classifying spaces of spinor groups, arXiv:0904.0800.
  72. [72]
    S.R. Edwards, On the spin bordism of b(e 8 × e 8), Illinois J. Math.35 (1991) 683.MathSciNetGoogle Scholar
  73. [73]
    L.E. Ibáñez and G.G. Ross, Discrete gauge symmetry anomalies, Phys. Lett.B 260 (1991) 291 [INSPIRE].ADSGoogle Scholar
  74. [74]
    L.E. Ibáñez, More about discrete gauge anomalies, Nucl. Phys.B 398 (1993) 301 [hep-ph/9210211] [INSPIRE].ADSMathSciNetGoogle Scholar
  75. [75]
    H.K. Dreiner, C. Luhn and M. Thormeier, What is the discrete gauge symmetry of the MSSM?, Phys. Rev.D 73 (2006) 075007 [hep-ph/0512163] [INSPIRE].ADSGoogle Scholar
  76. [76]
    R.N. Mohapatra and M. Ratz, Gauged discrete symmetries and proton stability, Phys. Rev.D 76 (2007) 095003 [arXiv:0707.4070] [INSPIRE].ADSGoogle Scholar
  77. [77]
    H.M. Lee et al., Discrete R symmetries for the MSSM and its singlet extensions, Nucl. Phys.B 850 (2011) 1 [arXiv:1102.3595] [INSPIRE].ADSzbMATHGoogle Scholar
  78. [78]
    H.P. Nilles, M. Ratz and P.K.S. Vaudrevange, Origin of family symmetries, Fortsch. Phys.61 (2013) 493 [arXiv:1204.2206] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  79. [79]
    M.-C. Chen et al., Anomaly-safe discrete groups, Phys. Lett.B 747 (2015) 22 [arXiv:1504.03470] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  80. [80]
    P.B. Gilkey and B. Botvinnik, The η invariant and the equivariant spin bordism of spherical space form 2 groups, Springer, Germany (1996).zbMATHGoogle Scholar
  81. [81]
    P.B. Gilkey, The η invariant of pin manifolds with cyclic fundamental groups, Period. Math. Hung.36 (1998) 139.MathSciNetzbMATHGoogle Scholar
  82. [82]
    L.E. Ibáñez and G.G. Ross, Discrete gauge symmetries and the origin of baryon and lepton number conservation in supersymmetric versions of the standard model, Nucl. Phys.B 368 (1992) 3 [INSPIRE].ADSMathSciNetGoogle Scholar
  83. [83]
    D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal and temperature, JHEP05 (2017) 091 [arXiv:1703.00501] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  84. [84]
    D. Gaiotto, Z. Komargodski and N. Seiberg, Time-reversal breaking in QCD 4, walls and dualities in 2 + 1 dimensions, JHEP01 (2018) 110 [arXiv:1708.06806] [INSPIRE].ADSzbMATHGoogle Scholar
  85. [85]
    J.J. van der Bij, A cosmotopological relation for a unified field theory, Phys. Rev.D 76 (2007) 121702 [arXiv:0708.4179] [INSPIRE].ADSGoogle Scholar
  86. [86]
    G.E. Volovik and M.A. Zubkov, Standard model as the topological material, New J. Phys.19 (2017) 015009 [arXiv:1608.07777] [INSPIRE].ADSGoogle Scholar
  87. [87]
    T.P. Cheng and L.F. Li, Gauge theory of elementary particle physics, Clarendon Press, Oxford U.K. (1984).Google Scholar
  88. [88]
    Particle Data Group collaboration, Review of particle physics, Chin. Phys.C 40 (2016) 100001 [INSPIRE].
  89. [89]
    L.N. Chang and C. Soo, The standard model with gravity couplings, Phys. Rev.C 53 (1996) 5682 [hep-th/9406188] [INSPIRE].ADSGoogle Scholar
  90. [90]
    C. Csáki, The minimal supersymmetric standard model (MSSM), Mod. Phys. Lett.A 11 (1996) 599 [hep-ph/9606414] [INSPIRE].ADSGoogle Scholar
  91. [91]
    F. Dillen and L. Verstraelen, Handbook of differential geometry, volume 1, Elsevier Science, The Netherlands (1999).Google Scholar
  92. [92]
    T. Banks and M. Dine, Note on discrete gauge anomalies, Phys. Rev.D 45 (1992) 1424 [hep-th/9109045] [INSPIRE].ADSMathSciNetGoogle Scholar
  93. [93]
    M. Berasaluce-González, M. Montero, A. Retolaza and A.M. Uranga, Discrete gauge symmetries from (closed string) tachyon condensation, JHEP11 (2013) 144 [arXiv:1305.6788] [INSPIRE].ADSGoogle Scholar
  94. [94]
    I. García-Etxebarria, M. Montero and A.M. Uranga, Closed tachyon solitons in type-II string theory, Fortsch. Phys.63 (2015) 571 [arXiv:1505.05510] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  95. [95]
    S. Hellerman, On the landscape of superstring theory in D > 10, hep-th/0405041 [INSPIRE].
  96. [96]
    K. Shiozaki, H. Shapourian and S. Ryu, Many-body topological invariants in fermionic symmetry-protected topological phases, Phys. Rev.B 95 (2017) 205139 [arXiv:1609.05970] [INSPIRE].ADSGoogle Scholar
  97. [97]
    R. Thom, Someglobalproperties of differentiable manifolds, World Scientific, Singapore (2012).Google Scholar
  98. [98]
    M.-C. Chen, M. Ratz and A. Trautner, Non-Abelian discrete R symmetries, JHEP09 (2013) 096 [arXiv:1306.5112] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  99. [99]
    T. Banks and N. Seiberg, Symmetries and strings in field theory and gravity, Phys. Rev.D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].ADSGoogle Scholar
  100. [100]
    M. Berasaluce-Gonzalez et al., Non-Abelian discrete gauge symmetries in 4d string models, JHEP09 (2012) 059 [arXiv:1206.2383] [INSPIRE].ADSMathSciNetGoogle Scholar
  101. [101]
    J. Polchinski, Monopoles, duality and string theory, Int. J. Mod. Phys.A 19S1 (2004) 145 [hep-th/0304042] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  102. [102]
    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP02 (2015) 172 [arXiv:1412.5148] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  103. [103]
    Y. Tachikawa and K. Yonekura, On time-reversal anomaly of 2 + 1d topological phases, PTEP2017 (2017) 033B04 [arXiv:1610.07010] [INSPIRE].Google Scholar
  104. [104]
    E. Witten, Five-brane effective action in M-theory, J. Geom. Phys.22 (1997) 103 [hep-th/9610234] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  105. [105]
    D.S. Freed and G.W. Moore, Setting the quantum integrand of M-theory, Commun. Math. Phys.263 (2006) 89 [hep-th/0409135] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  106. [106]
    A. Hattori, Integral characteristic numbers for weakly almost complex manifolds, Topology5 (1966) 259.MathSciNetzbMATHGoogle Scholar
  107. [107]
    R. Stong, Relations among characteristic numbersI, Topology4 (1965) 267.MathSciNetzbMATHGoogle Scholar
  108. [108]
    M.J. Hopkins and M.A. Hovey, Spin cobordism determines real K-theory, Math. Zeit.210 (1992) 181.MathSciNetzbMATHGoogle Scholar
  109. [109]
    O. Bergman, D. Rodríguez-Gómez and G. Zafrir, Discrete θ and the 5d superconformal index, JHEP01 (2014) 079 [arXiv:1310.2150] [INSPIRE].ADSGoogle Scholar
  110. [110]
    S. Sethi, A new string in ten dimensions?, JHEP09 (2013) 149 [arXiv:1304.1551] [INSPIRE].ADSMathSciNetGoogle Scholar
  111. [111]
    A. Dold and H. Whitney, Classification of oriented sphere bundles over a 4-complex, Annals Math.69 (1959) 667.MathSciNetzbMATHGoogle Scholar
  112. [112]
    L.M. Woodward, The classification of orientable vector bundles over CW-complexes of small dimension, Proc. Roy. Soc. EdinburghA 92 (1982) 175.MathSciNetzbMATHGoogle Scholar
  113. [113]
    R.C. Kirby and L.R. Taylor, A calculation of Pin+bordism groups, Comm. Math. Helv.65 (1990) 434.MathSciNetzbMATHGoogle Scholar
  114. [114]
    D.W. Anderson, E.H. Brown and F.P. Peterson, Pin cobordism and related topics, Comm. Math. Helv.44 (1969) 462.MathSciNetzbMATHGoogle Scholar
  115. [115]
    S.P. Novikov, Homotopy properties of Thom complexes, World Scientific, Singapore (2012).Google Scholar
  116. [116]
    W. Barth, K. Hulek, C. Peters and A. van de Ven, Compact complex surfaces, Series of Modern Surveys in Mathematics, Springer, Germany (2015).Google Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesDurham UniversityDurhamUnited Kingdom
  2. 2.Max-Planck-Institut für PhysikMünchenGermany
  3. 3.Instituut voor Theoretische FysicaKU LeuvenLeuvenBelgium
  4. 4.ITFUtrecht UniversityUtrechtNetherlands

Personalised recommendations