Where We Stand Today

  • Joseph Kouneiher


In his work of 1918, Hermann Weyl extended the general theory of relativity, which Albert Einstein had set forth in the years 1915–1916, to unify the two field phenomena known at this time, namely those described by electromagnetic and gravitational fields. But more was at stake. At the beginning of the paper in which Weyl worked out the mathematical foundations of the theory, he observed.



First of all, I apologize to those whose works I forgot to mention and which helped to improve the quality of the paper. Secondly, I would like to thank all my friends who have contributed to this volume, and all the colleagues who, through exchanges, suggestions or their own writings, have enriched the content of this paper. I am particularly grateful to Michael Atiyah, Alain Connes, Edward Witten, Roger Penrose, Misha Gromov, Ali Chamsddine, Lee Somlin, Jeremy Butterfield and John Stachel for exchanges and suggestions.


  1. 1.
    A. Ashtekar, New variables for classical and quantum gravity. Phys. Rev. Lett. 57, 2244 (1986)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Ashtekar, Gravity, Geometry and the Quantum, in Vers une nouvelle Philosophie de la nature, Joseph Kouneiher ed. Hermann, 2010Google Scholar
  3. 3.
    A. Ashtekar, J. Lewandowski, Quantum theory of geometry. I: area operators. Class. Quant. Grav.14 (1997) A55–A82. Scholar
  4. 4.
    A. Ashtekar, J. Lewandowski, Quantum theory of geometry. II: volume operators. Adv. Theor. Math. Phys.1 (1998) 388. Scholar
  5. 5.
    M.F. Atiyah, I.M. Singer, The index of elliptic operators on compact manifolds. Bull. Amer. Math. Soc. 69(3), 422–433 (1963)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M.F. Atiyah, I.M. Singer, The index of elliptic operators I. Ann. Math. 87(3), 484–530 (1968)MathSciNetCrossRefGoogle Scholar
  7. 7.
    M.F. Atiyah, I.M. Singer, The index of elliptic operators V. Ann. Math. Second Ser. 93(1), 139–149 (1971)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Atiyah, R. Dijkgraaf, N.l Hitchin, Geometry and physics. Phil. Trans. R. Soc. A 368, 913–926 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Atiyah, N.S. Manton, B.J. Schroers, Geometric Models of Matter, arXiv:1108.5151
  10. 10.
    K.A. Brading, T.A. Ryckman, Hilbert’s foundations of physics’: gravitation and electromagnetism within the axiomatic method. Stud. Hist. Philos. Sci. B: Stud. Hist. Philos. Mod. Phys. 39(1), 102–153 (2008)Google Scholar
  11. 11.
    N.N. Bogoliubov, D.V. Shirkov, The Theory of Quantized Fields (Interscience, New York, 1959)zbMATHGoogle Scholar
  12. 12.
    N.N. Bogoliubov, O. Parasiuk, On the multiplication of the causal function in the quantum theory of fields. Acta Math. 97, 227–266 (1957)MathSciNetCrossRefGoogle Scholar
  13. 13.
    R. Bott, On Mathematics and Physics, Collected Works, vol. 4, p. 382CrossRefGoogle Scholar
  14. 14.
    K. Brading, E. Castellani, Symmetries in Physics: Philosophical Reflections, 2003Google Scholar
  15. 15.
    P. Candelas, P. Green, L. Parke, X. de la Ossa, A pair of Calabi-Yau manifoldsas an exactly soluble superconformal eld theory. Nucl. Phys. B 359, 21–74 (1991)
  16. 16.
    A.H. Chamseddine, A. Connes, M. Marcolli, Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    S. Chern, J. Simons, Some cohomology classes in principal fiber bundles and their application to Riemannian geometry, Proc. Nat. Acad. Sci. USA 68, 791–794, Or, characteristic forms and geometrical invariants. Ann. Math. 99(48–69), 1974 (1971)MathSciNetCrossRefGoogle Scholar
  18. 18.
    S. Chern, Vector bundles with a connection, Studies in Global Differential Geometry. Math. Asso. Amer. Studies No. 27, 1–26 (1989)Google Scholar
  19. 19.
    S. Chern, Complex Manifolds without Potential Theory, 2nd edn. (Springer, Berlin, 1979)CrossRefGoogle Scholar
  20. 20.
    S.-S. Chern, What Is Geometry? The American Mathematical Monthly, vol. 97(8), Special Geometry Issue, pp. 679-686 (1990)MathSciNetCrossRefGoogle Scholar
  21. 21.
    S. Coleman, Quantum sine-Gordon equation as the massive Thirring model. Phys. Rev. D 11, 2088 (1975)CrossRefGoogle Scholar
  22. 22.
    A. Connes, M. Marcolli, Noncommutative Geom. (American Mathematical Society, Quantum Fields and Motives, 2007)Google Scholar
  23. 23.
    Alain Connes, Dirk Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem II: the \(\beta \)-function, diffeomorphisms and renormalization group. Commun. Math. Phys. 216, 215–241 (2001). arXiv:hep-th/0003188v1MathSciNetCrossRefGoogle Scholar
  24. 24.
    A. Connes, M. Marcolli, Renormalization, the Riemann-Hilbert correspondence and motivic Galois theory, Frontiers in number theory, physics and geometry, vol. II (Springer, Berlin, 2007). pp. 617–713Google Scholar
  25. 25.
    A. Connes, Noncommutative Geometry (Academic Press, Cambridge, 1994)zbMATHGoogle Scholar
  26. 26.
    A. Connes, J. Lott, Particle models and noncommutative geometry. Nucl. Phys. Proc. Suppl. B18, 29 (1989)MathSciNetzbMATHGoogle Scholar
  27. 27.
    A. Connes Geometry and PhysicsGoogle Scholar
  28. 28.
    A. Connes, D. Kreimer, Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199, 203 (1998). arXiv:hep-th/9808042MathSciNetCrossRefGoogle Scholar
  29. 29.
    A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I: the Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210, 249 (2000). arXiv:hep-th/9912092MathSciNetCrossRefGoogle Scholar
  30. 30.
    A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II: the beta-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216, 215 (2001). arXiv:hep-th/0003188CrossRefGoogle Scholar
  31. 31.
    A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives. preliminary version available at
  32. 32.
    A. Connes, D. Kreimer, Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys. 199, 203–242 (1998)MathSciNetCrossRefGoogle Scholar
  33. 33.
    A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Comm. Math. Phys. 210(1), 249–273 (2000)MathSciNetCrossRefGoogle Scholar
  34. 34.
    A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The \(\beta \)-function, diffeomorphisms and the renormalization group. Comm. Math. Phys. 216(1), 215–241 (2001)MathSciNetCrossRefGoogle Scholar
  35. 35.
    A. Connes, M. Marcolli, From Physics to Number theory via Noncommutative Geometry, II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory, to appear in Frontiers in Number Theory, Physics, and Geometry, vol. II. Preprint hep-th/0411114Google Scholar
  36. 36.
    D. A. Cox, S. Katz, Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs no. 68. Providence, RI: American Mathematical Society. 1999Google Scholar
  37. 37.
    L. Corry, David Hilbert and the Axiomatization of Physics (1898-1918): From Grundlagen der Geometrie to Grundlagen der Physik (Kluwer Academic Publishers, Dordrecht, 2004). p. 429Google Scholar
  38. 38.
    B. Delamotte, A hint of renormalization. Am. J. Phys., 72(2) (2004)CrossRefGoogle Scholar
  39. 39.
    P. Deligne, Quelques idées maîtresses de l’œuvre de A. Grothendieck, Matériaux pour l’histoire des mathématiques au XXe sicle, in Proceedings of the workshop on the honour of Jean Dieudonné (Nice 1996), France Mathematical society, pp. 11–19 (1998)Google Scholar
  40. 40.
    R. Dijkgraaf, The mathematics of strings theory, Séminaire Poincaré, 2004Google Scholar
  41. 41.
    P. A. M. Dirac, Quantised singularities in the electromagnetic field. Proc. Roy. Soc. A 133, 60CrossRefGoogle Scholar
  42. 42.
    P.A.M. Dirac, The Relation between Mathematics and Physics. Proc. R. Soc. (Edinburgh) 59(Part II), 122–129 (1939)Google Scholar
  43. 43.
    S. Donaldson, P. Kronheimer, The Geometry of Four-Manifolds (Oxford, 1990)Google Scholar
  44. 44.
    S. Donaldson, J. Diff. Geom. 18, 269 (1983)CrossRefGoogle Scholar
  45. 45.
    S. Donaldson, R. Friedman, Connected sums of self-dual manifolds and deformations of singular spaces. Nonlinearity 2, 197–239 (1989)MathSciNetCrossRefGoogle Scholar
  46. 46.
    R. Durrer, R. Maartens, Dark energy and dark gravity, Gen. Rel. Grav. 40, 301–328 (2008) arXiv:0711.0077 (2007)
  47. 47.
    F.J. Dyson, The S-matrix in quantum electrodynamics. Phys. Rev. 75, 1736. (1949)CrossRefGoogle Scholar
  48. 48.
    G. Efstathiou, in The Physics of the Early Universe, ed. by J.A. Peacock, A.F. Heavens, A. Davies (Adam-Higler, Bristol, 1990)Google Scholar
  49. 49.
    A. Einstein, Philosopher-Scientist, in The Library of Living Philosophers, ed. by P.A. Schilpp (Evanston, 1949), pp. 2–95Google Scholar
  50. 50.
    A. Einstein, Ideas and opinions, quoted from Schweber, Einstein and Oppenheimer: the meaning of genius (1954)Google Scholar
  51. 51.
    A. Floer, Bull. Am. Math. Soc. 16, 279 (1987)MathSciNetCrossRefGoogle Scholar
  52. 52.
    A. Friedman, Über die Krümmung des Raumes. Z. Phys. 10(1), 377–386 (1922)CrossRefGoogle Scholar
  53. 53.
    A. Friedman, Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes. Z. Phys. 21(1), 326–332 (1924)CrossRefGoogle Scholar
  54. 54.
    D.S. Freed, G.W. Moore, Twisted equivariant matter. Ann. Henri Poincare 14, 1927 (2013). arXiv:1208.5055 [hep-th]MathSciNetCrossRefGoogle Scholar
  55. 55.
    F. Helein,Dualités, supersymétries et systèmes complètement intégrables, in Vers une nouvelle Philosophie de la nature, ed. by J. Kouneiher, Hermann edn. (2010)Google Scholar
  56. 56.
    F. Helein, J. Kouneiher, On the soliton-particle dualities, in Geometries of Nature, Living Systems and Human Cognition, ed. by L. Boi (World Scientific, 2005). pp. 93–128CrossRefGoogle Scholar
  57. 57.
    K. Hepp, Proof of the Bogoliubov-Parasiuk theorem on renormalization. Commun. Math. Phys. 2, 301–326 (1966)CrossRefGoogle Scholar
  58. 58.
    D. Hilbert, Uber die Grundlagen der Geometrie. Gottinger Nachrichten 233–241 (1902)Google Scholar
  59. 59.
    D. Hilbert, Lecture delivered before the International Congress of Mathematicians, Paris France (1900). English translation appeared in Bull. Am. Math. Soc. 8 (1902), 437–479. A reprint of appears in Mathematical Developments Arising from Hilbert Problems, ed. by F. Brouder, Am. Math. Soc. 1976. The original address “Mathematische Probleme” appeared in Göttinger Nachrichten, 1900, pp. 253–297, and in Archiv der Mathematik und Physik, 3(1), 44–63 and 213–237 (1901)Google Scholar
  60. 60.
    N. Hitchin, Interaction between mathematics and physics, ARBOR Ciencia, Pensamiento y Cultura, CLXXXIII 725, mayo-junio, pp. 427–432 (2007)Google Scholar
  61. 61.
    G. ’t Hooft, M. Veltman, Regularization and renormalization of gauge fields. Nucl. Phys. B 44, 189 (1972)MathSciNetCrossRefGoogle Scholar
  62. 62.
    K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow, Mirror symmetry. Clay Mathematics Monographs. Providence, RI: American Mathematical Society (2003)Google Scholar
  63. 63.
    E. Hubble, A relation between distance and Radial Velocity among Extra-Galactic Nebulae. Proc. Natl. Acad. Sci. 15, 168–173 (1929)CrossRefGoogle Scholar
  64. 64.
    V.F.R. Jones, A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. 12, 103–111 (1985). Scholar
  65. 65.
    F. Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen, in Gesammelte mathematische Abhandlungen, vol. i (1872) pp. 460–497CrossRefGoogle Scholar
  66. 66.
    F. Klein, Vorlesungen über die Entwicklung der Mathematik im 19 (Wissenschaftliche Buchgesellschaft, Jahrhundert. Darmstadt, 1979)CrossRefGoogle Scholar
  67. 67.
    M. Kontsevich, Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435 [math.AG]
  68. 68.
    Kontsevich, M. Homological algebra of mirror symmetry. In Proc. Int. Congress of Mathematicians, Zrich, vols. 1-2 (Birkhuser, Basel, Switzerland, 1995). pp. 120–139CrossRefGoogle Scholar
  69. 69.
    M. Kontsevich, M. Yu, Gromov-Witten classes, quantum cohomology, and enumerativegeometry. Commun. Math. Phys. 164, 525–562 (1994). Scholar
  70. 70.
    J. Kouneiher, in Leibniz and the Dialogue between Sciences, Philosophy and Engineering, 1646-2016. New Historical and Epistemological Insights, ed. by R. Pisano, M. Fichant, P. Bussotti, A.R.E. Oliveira (The College’s Publications, London, 2017)Google Scholar
  71. 71.
    J.Kouneiher, Conceptual Foundations of Soliton Versus Particle Dualities Toward a Topological Model for Matter. International Journal of Theoretical Physics, vol. 55(6), pp. 2949–2968. 20p (2016)CrossRefGoogle Scholar
  72. 72.
    J. Kouneiher, C. Barbachoux, Cartan’s soldered spaces and conservation laws in physics. Int. J. Geom. Methods Mod. Phys. 12(9) (2015)MathSciNetCrossRefGoogle Scholar
  73. 73.
    J. Kouneiher, Geometric Continuum and the Birth of the Mathematical-Physics, to appear in IJGMMP, 2018Google Scholar
  74. 74.
    T. Krajewski, P. Martinetti, Wilsonian renormalization, differential equations and Hopf algebras, Talk given by T. Krajewski at the conference “Combinatorics and Physics" Max Planck Institut Für Mathematik Bonn, March 2007Google Scholar
  75. 75.
    D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2, 303 (1998). arXiv:q-alg/9707029MathSciNetCrossRefGoogle Scholar
  76. 76.
    D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2, 303–334 (1998)MathSciNetCrossRefGoogle Scholar
  77. 77.
    G. Lazardies, Introduction to Cosmology, arXiv:hep-ph/9904502
  78. 78.
    G. Lemaitre, Ann. Soc. Sci. Brux. A53, 81 (1933)Google Scholar
  79. 79.
    Colin MacLarty, How Grothendieck simplified algebraic geometry. Not. AMS 63(3), 250 (2016)MathSciNetGoogle Scholar
  80. 80.
    N.S. Manton, Skyrme fields and instantons, in "Geometry of Lowdimensional Manifolds : 1", ed. by S.K. Donaldson, C.B. Thomas, Lond. Math. Soc. Lec. Notes Ser. 150 (Cambridge University Press, Cambridge, 1990)Google Scholar
  81. 81.
    S. Mandelstam, Soliton operators for the quantized sine-Gordon equation. Phys. Rev. D 11(10), 3026–30 (1975)MathSciNetCrossRefGoogle Scholar
  82. 82.
    Y.I. Manin, M. Marcolli, Big Bang, blowup, and modular curves: algebraic geometry in cosmology. Symmetry Integrability Geom.: Methods Appl. SIGMA 10, 73 (2014)MathSciNetzbMATHGoogle Scholar
  83. 83.
    J.C. Maxwell, Mathematical and Physical Science, Section A (Mathematical and Physical Sciences) of the British Association, Liverpool, 1870. Nature 2, 419–422 (1870)CrossRefGoogle Scholar
  84. 84.
    T. Miwa, M. Jimbo, E. Date, Solitons, Cambridge Tracts in Math. 135 (Cambridge University Press, Cambridge, 2000)Google Scholar
  85. 85.
    C. Montonen, D. Olive, Magnetic monopoles as gauge particles, Phys. Lett. 72B(1) (1977)CrossRefGoogle Scholar
  86. 86.
    G.W. Moore, Physical Mathematics and the Future, PreprintGoogle Scholar
  87. 87.
    J. von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren. Math. Ann. 104, 570 (1931)MathSciNetCrossRefGoogle Scholar
  88. 88.
    E.T. Newman, A fundamental solution to the CCC equations, Gen. Relativ. Gravit. 46(5), 1717, 13p (2014). arXiv:1309.7271
  89. 89.
    H. Nicolai, K. Peeters, M. Zamaklar, Loop Quantum Gravity: An Outside View, 2014. arXiv:hep-th/0501114
  90. 90.
    D. Olive, E. Witten, Supersymmetry algebra that include topological charges, Phys. Lett. 78B(1) (1978)Google Scholar
  91. 91.
    P.J.E., Peebles, The Large-scale Structure of the Universe (Princeton University Press, Princeton, 1980)Google Scholar
  92. 92.
    P.J.E. Peebles, D.N. Schramm, E.L. Turner, R.G. Kron, Nature 352, 769 (1991)CrossRefGoogle Scholar
  93. 93.
    Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe (A.A. Knopf, New York, 2005)zbMATHGoogle Scholar
  94. 94.
    R. Penrose, Angular momentum: an approach to combinatorial space-time, in Quantum Theory And Beyond, ed. by T. Bastin (Cambridge University Press, Cambridge, 1971)Google Scholar
  95. 95.
    R. Penrose, W. Rindler, Spinors and space-time, in Two-Spinor Calculus and Relativitic Fields, vol. 1 (Cambridge University Press, Cambridge, 1984)Google Scholar
  96. 96.
    H. Poincaré, Sur les rapport de l’analyse pure et de la physique mathématique, Address to the 1897 ICM, ZurichGoogle Scholar
  97. 97.
    H. Poincaré, Analysis situs. J. Sec. Polyt. 1, 1–121 (1895)zbMATHGoogle Scholar
  98. 98.
    H. Poincaré, Sur la connexion des surfaces algébriques. C. R. Acad. Sc. 133, 969–973 (1901)zbMATHGoogle Scholar
  99. 99.
    H. Poincaré, Sur les cycles des surfaces algébriques; quatrième complement a l’Analysis situs. J. Math. Pures Appl. 8, 169–214 (1902)Google Scholar
  100. 100.
    A. Pressley, G.B. Segal, Loop Groups (Oxford University Press, Oxford, 1986)zbMATHGoogle Scholar
  101. 101.
    H.P. Robertson, On the foundations of relativistic cosmology. Proc. Natl. Acad. Sci. 15(11), 822–829 (1929)CrossRefGoogle Scholar
  102. 102.
    I. Robinson, Report to the Eddington Group, Cambridge, (1956)Google Scholar
  103. 103.
    I. Robinson, J. Math. Phys. 2, 290 (1961)CrossRefGoogle Scholar
  104. 104.
    C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, 2010)zbMATHGoogle Scholar
  105. 105.
    C. Rovelli, L. Smolin, Loop space representation of quantum general relativity. Nucl. Phys. B 331, 80 (1990)MathSciNetCrossRefGoogle Scholar
  106. 106.
    J. Rosenberg, A selective history of the Stone-von Neumann theorem’, in Operator Algebras, Quantization, and Noncommutative Geometry: A Centennial Celebration Honoring John von Neumann and Marshall H. Stone, ed. by R.S. Doran, R.V. Kadison, Contemporary Mathematics, vol. 365 (American Mathematical Society, 2004)Google Scholar
  107. 107.
    Y. Ruan, G. Tian, A mathematical theory of quantum cohomology. J. Differ. Geom. 42, 259–367 (1995). Scholar
  108. 108.
    T. Sauer, The Relativity of Discovery: Hilbert’s First Note on the Foundations of Physics. arXiv:physics/9811050
  109. 109.
    U. Majer, T. Sauer, Hilbert’s World Equations and His Vision of a Unified Science.
  110. 110.
    S.S. Schweber, Qed and the Men Who Made It: Dyson (Schwinger, and Tomonaga, Princeton University Press, Feynman, 1994)zbMATHGoogle Scholar
  111. 111.
    A. Sen, Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and SL(2, Z ) invariance in string theory. Phys. Lett. B 329, 217–221 (1994)MathSciNetCrossRefGoogle Scholar
  112. 112.
    N. Seiberg, E. Witten, Monopole condensation, and connement in N = 2 supersymmetricYang-Mills theory. Nucl. Phys. B 426, 19–52 (1994)CrossRefGoogle Scholar
  113. 113.
    T.R.H. Skyrme, A Unified Theory for Mesons and Baryons. Nucl. Phys. 31, 556 (1962); Proc. Roy. Soc. A 247, 260 (1958)Google Scholar
  114. 114.
    T.R.H. Skyrme, Kinks and the Dirac equation. J. Math. Phys. 12, 1735–42 (1971)MathSciNetCrossRefGoogle Scholar
  115. 115.
    T.H.R. Skyrme, A non-linear field theory. Proc. Roy. Soc. A260, 127–138 (1961)MathSciNetCrossRefGoogle Scholar
  116. 116.
    C. Rovelli, L. Smolin, Spin networks and quantum gravity. Phys. Rev. D 52, 5743–5759 (1995)MathSciNetCrossRefGoogle Scholar
  117. 117.
    H. Spiesberger, M. Spira, P.M. Zerwas. The Standard Model: Physical Basis and Scattering Experiments. Appears in: Scattering, ed. by R. Pike et al., vol. 2, (Academic Press, Cambridge, 2002). 1505–1533CrossRefGoogle Scholar
  118. 118.
    M.H. Stone, Linear transformations in Hilbert space, III: operational methods and group theory. Proc. Nat. Acad. Sci. 16, 172–175 (1930)CrossRefGoogle Scholar
  119. 119.
    J.J. Sylvester, A plea for the mathematician, II. Nature 1(10), 261–263 (1870)CrossRefGoogle Scholar
  120. 120.
    W. Thirring, Ann. Phys. (N.Y.) 3, 91 (1958)CrossRefGoogle Scholar
  121. 121.
    P. Tod, Penrose’s circles in the CMB and a test of inflation. Gen. Relativ. Gravitat. 44, 2933–2938 (2012). arXiv:1107.1421MathSciNetCrossRefGoogle Scholar
  122. 122.
    R. Vakil, Foundations of Algebraic Geometry.
  123. 123.
    A.G. Walker, On Milne’s theory of world-structure. Proc. Lond. Math. Soc., Ser. 2, 42(1), 90–127 (1937)Google Scholar
  124. 124.
    H. Weyl, Reine Infinitesimalgeometrie, in Weyl, Gesammelte Abhandlungen, 4 vols., vol. II (Springer, Berlin, 1968). pp. 1-28, on p. 2Google Scholar
  125. 125.
    E. Witten, Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)MathSciNetCrossRefGoogle Scholar
  126. 126.
    E. Witten, Quantum eld theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989)CrossRefGoogle Scholar
  127. 127.
    E. Witten, Monopoles and four manifolds. Math. Res. Lett. 1, 769–796 (1994)MathSciNetCrossRefGoogle Scholar
  128. 128.
    E. Witten, Quantum field theory and the Jones polynomial, Braid Group, Knot Group, and Statistical Mechanics, ed. by C. N. Yang, M. L. Ke (World Scientific, 1989). pp. 239–329CrossRefGoogle Scholar
  129. 129.
    E. Witten, Topological quantum field theory. Comm. Math. Phys. 117(3), 353–386 (1988)MathSciNetCrossRefGoogle Scholar
  130. 130.
    K.G. Wilson, The renormalization group: critical phenomena and the Kondo problem. Rev. Mod. Phys. 47(4), 773 (1975)MathSciNetCrossRefGoogle Scholar
  131. 131.
    C.N. Yang, Magnetic monopoles, fiber bundles, and gauge fields. Ann. NY Acad. Sci. 294, 86–97 (1977)CrossRefGoogle Scholar
  132. 132.
    C.N. Yang, R.L. Mills, Conservation of isotopic spin and isotopic gauge invariance. Phys. Rev. 96, 191–195 (1954)MathSciNetCrossRefGoogle Scholar
  133. 133.
    W. Zimmermann, Convergence of Bogoliubov’s method of renormalization in momentum space’. Commun. Math. Phys. 15, 208 (1968)MathSciNetCrossRefGoogle Scholar
  134. 134.
    J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford University Press, Oxford, 1999)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Côte d’Azur University and Lab. ARTEMIS UMR 7250 (OCA, UCA, CNRS)NiceFrance
  2. 2.Nice and Sophia Antipolis UniversityNiceFrance

Personalised recommendations