New string amplitudes from old fermat (hyper)surfaces

International Journal of Modern Physics A

Volumn 19, Issue 11, 2004, Pages 1655-1703

  Kholodenko, A L ^a  

^a CLEMSON UNIVERSITY   (United States)

Author keywords

Fermat hypersurfaces; Hodge spectrum; Member theoretic zeta functions; Picard Fuchs equations; Singularity theory; Veneziano amplitudes

Indexed keywords

AMPLITUDE MODULATION; CALCULATION; GEOMETRY; MASS SPECTROMETRY; MATHEMATICAL ANALYSIS; MATHEMATICAL COMPUTING; REVIEW;

EID: 14244266337     PISSN: 0217751X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0217751X04018166     Document Type: Review

References (117)

1. G. Veneziano, Il Nuovo Cimento A57, 190 (1968).
2. V. De Alfaro, S. Fubini, G. Furlan and C. Rossetti, Currents in Hadron Physics (Elsevier, Amsterdam, 1973).
3. M. Virasoro, Phys. Rev. 177, 2309 (1969).
4. A. Kholodenko, J. Geom. Phys. 38, 81 (2001).
5. A. Kholodenko, J. Geom. Phys. 33, 59 (2000).
6. A. Kholodenko, J. Geom. Phys. 33, 23 (2000).
7. M. Jacob, Dual Theory (Elsevier, Amsterdam, 1974).
8. W. Heisenberg, Z. Phys. 120, 513 (1943); ibid. 120, 673 (1943).
9. W. Heisenberg, Z. Phys. 120, 513 (1943); ibid. 120, 673 (1943).
10. J. Bjorken and S. Drell, Relativistic Quantum Fields, Vol. 2 (McGraw-Hill, New York, 1965).
11. S. Novikov, S. Manakov, L. Pitaevskii and V. Zakhaxov, Theory of Solitons (Consultants Bureau, New York, 1984).
12. D. Anosov, S. Aranson, V. Arnol'd, I. Bronstein, V. Grines and Yu. Il'yashenko, Ordinary Differential Equations and Smooth Dynamical Systems (Springer-Verlag, Berlin, 1997).
13. L. Landau, Nucl. Phys. 13, 181 (1959).
14. F. Pham, Bull. Soc. Math. France 93, 333 (1965).
15. J. Milnor, Singular Points of Complex Hypersurfaces (Princeton University Press, Princeton, New Jersey, 1968).
16. E. Brieskorn, Inv. Math. 2, 1 (1966).
17. G. Ponzano, T. Regge, E. Speer and M. Westwater, Commun. Math. Phys. 15, 83 (1969); ibid. 18, 1 (1970).
18. G. Ponzano, T. Regge, E. Speer and M. Westwater, Commun. Math. Phys. 15, 83 (1969); ibid. 18, 1 (1970).
19. M. Green, J. Schwarz and E. Witten, Supersting Theory, Vol. 1 (Cambridge University Press, Cambridge, UK, 1987).
20. J. Edwards, A Treatise on the Integral Calculus, Vol. 2 (Macmillan, London, UK, 1922).
21. A. Varchenko, Sov. Math. Doklady 53, 1206 (1989); ibid. 54, 146 (1990).
22. A. Varchenko, Sov. Math. Doklady 53, 1206 (1989); ibid. 54, 146 (1990).
23. S. Chowla and A. Selberg, J. Für die Reine und Angew. Math. 227, 86 (1967).
24. A. Weil, Commun. Pure Appl. Math. 29, 813 (1976).
25. A. Weil, Abelian Varieties and the Hodge Ring, Collected Papers, Vol. 3 (Springer-Verlag, Berlin, 1979).
26. B. Gross, Inv. Math. 45, 193 (1978).
27. S. Lang, Introduction to Algebraic and Abelian Functions (Springer-Verlag, Berlin, 1982).
28. V. Arnol'd, S. Gusein-Zade and A. Varchenko, Singularities of Differentiable Maps, Vol. 2 (Birkhäuser, Boston, Massachusetts, 1988).
29. P. Deligne, LNM 900, 8 (1982).
30. V. Vladimirov, I. Volovich and E. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).
31. E. Artin, Algebraic Numbers and Algebraic Functions (Gordon and Breach, London, UK, 1967).
32. D. Goldschmidt, Algebraic Functions and Projective Curves (Springer-Verlag, Berlin, 2003).
33. M. Rosen, Number Theory in Function Fields (Springer-Verlag, Berlin, 2002).
34. W. Gilbert, Modern Algebra with Applications (John Wiley & Sons, New York, 1976).
35. H. Stichtenoth, Algebraic Function Fields and Codes (Springer-Verlag, Berlin, 1993).
36. S. Lang, Undergraduate Algebra (Springer-Verlag, Berlin, 1990).
37. B. Van der Waerden, Algebra (Springer-Verlag, Berlin, 1967).
38. B. Mazur, AMS Bulletin 29, 14 (1993).
39. L. Childs, A Concrete Introduction to Higher Algebra (Springer-Verlag, Berlin, 1995).
40. W. Li, Number Theory with Applications (World Scientific, Singapore, 1996).
41. R. Bott, Adv. Math. 16, 144 (1975).
42. Y. Aurby and M. Perret, A Weil theorem for singular curves, in Arithmetic, Geometry and Coding Theory, eds. R. Pellikaan, M. Perret and S. Vladut (Walter de Gruyter, Berlin, 1996).
43. V. Guillemin and A. Pollack, Differential Topology (Prentice Hall, New York, 1974).
44. J. Milnor, Infinite Cyclic Coverings, Collected Papers, Vol. 2 (Publish or Perish, Inc., Houston, TX, 1995).
45. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, UK, 1995).
46. J. Birman and M. Kidwell, Adv. Math. 46, 217 (1982).
47. N. Yui, Arithmetic of certain Calabi-Yau varieties and mirror symmetry, in Arithmetic Algebraic Geometry, eds. B. Conrad and K. Rubin (AMS Publications, Providence, RI, 2001).
48. D. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry (AMS Publications, Providence, RI, 1999).
49. L. Kauffman, On Knots (Princeton University Press, Princeton, New Jersey, 1987).
50. G. Brude and H. Zieschang, Knots (Walter de Gryter, Berlin, 1985).
51. M. Armstrong, Basic Topology (Springer-Verlag, Berlin, 1983).
52. D. Rolfsen, Knots and Links (Publish or Perish, Houston, TX, 1990).
53. A. Grothendieck, LNM 288, 5 (1972).
54. A. Wiles, Ann. Math. 141, 143 (1995).
55. A. Kholodenko, J. Geom. Phys. 43, 45 (2002).
56. A. Weil, AMS Bulletin 55, 497 (1949).
57. R. Miranda, Algebraic Curves and Riemann Surfaces (AMS Publications, Providence, RI, 1997).
58. D. Rohrlich, Inv. Math. 39, 95 (1977).
59. B. Gross and N. Koblitz, Ann. Math. 109, 569 (1979).
60. N. Koblitz, p-Adic Analysis, A Short Course on Recent Work (Cambridge University Press, Cambridge, UK, 1980).
61. L. Landau and E. Lifshitz, Mechanics (Nauka Publishers, Moscow, 1965).
62. G. Frey, Ann. Univ. Saraviensis, Ser. Math. 1, 1 (1986).
63. H. McKean and V. Moll, Elliptic Curves (Cambidge University Press, Cambridge, UK, 1999).
64. S. Lang, Cyclotomic Fields (Springer-Verlag, Berlin, 1990).
65. L. Washington, Introduction to Cyclotomic Fields (Springer-Verlag, Berlin, 1997).
66. M. Mulase, Algebraic theory of the KP equations, in Perspectives in Mathematical Physics (International Press, Cambridge, Massachusetts, 1994).
67. A. Kholodenko, J. Geom. Phys. 35, 193 (2000).
68. P. Griffiths, Ann. Math. 90, 460 (1969); ibid. 90, 496 (1969).
69. P. Griffiths, Ann. Math. 90, 460 (1969); ibid. 90, 496 (1969).
70. Y. Manin, AMS Translations 37, 59 (1964).
71. Y. Shimizu and K. Ueno, AMS Translations 206, 1 (2002).
72. D. Husemöller, Elliptic Curves (Springer-Verlag, Berlin, 1987).
73. J. Silverman, The Arithmetic of Elliptic Curves (Springer-Verlag, Berlin, 1986).
74. V. Kulikov, Mixed Hodge Structures and Singularities (Cambridge University Press, Cambridge, UK, 1998).
75. S. Lefshetz, Applications of Algebraic Topology to Graphs and Networks, The Picard-Lefshetz Theory and Feynman Integrals (Springer-Verlag, Berlin, 1975).
76. V. Schechtman and A. Varchenko, Inv. Math. 106, 139 (1991).
77. E. Brieskorn and H. Knorrer, Plane Algebraic Curves (Birkhäuser, Boston, 1986).
78. R. Wells, Differential Analysis on Complex Manifolds (Prentice Hall, New Jersey, 1973).
79. A. Weil, Introduction a L'etude des Varietes Kähleriennes (Hermann, Paris, 1958).
80. V. Arnol'd, Mathematical Methods of Classical Mechanics (Nauka Publishers, Moscow, 1974).
81. V. Arnol'd and A. Givental, Symplectic geometry, in Dynamical Systems IV, eds. V. Arnol'd and S. Novikov (Springer-Verlag, Berlin, 1990).
82. A. Varchenko, Math. USSR, Izvestija 18, 469 (1982).
83. P. Griffiths, Am. J. Math. 90, 568 (1968).
84. P. Griffiths, Am. J. Math. 90, 805 (1968).
85. R. Hwa and V. Teplitz, Homology and Feynman Integrals (W. A. Benjamin, Inc., New York, 1966).
86. D. Eisenbud and W. Neumann, Three-Dimensional Link Theory and Invariants of Plane Curve Singularities (Princeton University Press, Princeton, New Jersey, 1985).
87. L. Trang, Compositio Math. 25, 281 (1972).
88. D. Sumners and J. Woods, Inv. Math. 40, 107 (1977).
89. N. Katz, Seminaire Bourbaki 409, 167 (1972).
90. J. Franks, Homology and Dynamical Systems, Regional Conf. Series in Math. 49 (AMS Publications, Providence, RI, 1982).
91. M. Sebastiani and R. Thom, Inv. Math. 13, 90 (1971).
92. M. Oka, Topology 12, 19 (1973).
93. N. Saveliev, Lectures on the Topology of 3-Manifolds (Walter de Gruyter, Berlin, 1999).
94. N. A'Campo, Commun. Math. Helv. 50, 233 (1975).
95. K. Murasugi, Knot Theory and its Applications (Birkhäuser, Boston, 1996).
96. A. Kawauchi, A Survey of Knot Theory (Birkhäuser, Boston, 1996).
97. I. Gelfand, M. Kapranov and A. Zelevinski, Discriminants, Resultants and Multidimensional Determinants (Birkhäuser, Boston, 1994).
98. B. Buchstaber and T. Panov, Torus Actions and their Applications in Topology and Combinatorics, Univ. Lect. Series 24 (AMS Publications, Providence, RI, 2002).
99. V. Arnol'd, V. Vaasil'ev, V. Goryunov and O. Lyashko, Singularities, local and global theory, in Dynamical Systems VI, ed. V. Arnold (Springer-Verlag, Berlin, 1993).
100. T. Zaslavsky, Facing up to Arrangements, Face-Count Formulas for Partitions of Space by Hyperplanes, AMS Memoirs 154 (AMS Publications, Providence, RI, 1975).
101. P. Orlik and H. Terao, Arrangements and Hypergeometric Integrals, Math. Sci. Japan Memoirs 9 (Tokyo, 2001).
102. P. Etingof, I. Frenkel and A. Kirillov, Jr., Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations (AMS Publications, Providence, RI, 1998).
103. A. Varchenko, Special functions, KZ type equations and representation theory, Math.QA/0205313.
104. V. Arnol'd, Funct. Anal. Appl. 12, 1 (1978).
105. V. Batyrev, Duke Math. J. 69, 349 (1993).
106. A. Koushnirenko, Inv. Math. 32, 1 (1976).
107. S. Salamon, Topology 35, 137 (1996).
108. V. Danilov and A. Khovanskii, Math. USSR - Izv. 29, 279 (1987).
109. D. Cox, Recent developments in toric geometry, Math.AG/9606016.
110. J. Steenbrink, Mixed Hodge Structure on Vanishing Cohomology (Proc. Nordic Summer School, Oslo, 1976).
111. M. Saito, Math. Ann. 289, 703 (1991).
112. A. Varchenko and A. Khovanskii, Sov. Math. Dokl. 32, 122 (1985).
113. A. Borel, Linear Algebraic Groups (W. A. Benjamin, New York, 1969).
114. I. Macdonald, Lectures on Lie Groups and Lie Algebras (Cambridge University Press, Cambridge, UK, 1995).
115. M. Atiyah, Proc. Edinburg Math. Soc. 26, 121 (1983).
116. W. Fulton and J. Harris, Representation Theory (Springer-Verlag, Berlin, 1991).
117. V. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge University Press, Cambridge, UK, 1984).