String partition functions and infinite products

Advances in Theoretical and Mathematical Physics

Volumn 4, Issue 2, 2000, Pages 397-485

  Kawai, Toshiya ^a   Yoshioka, Kota ^b  

^a KYOTO UNIVERSITY   (Japan)

^b KOBE UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001618740     PISSN: 10950761     EISSN: 10950753     Source Type: Journal    
DOI: 10.4310/atmp.2000.v4.n2.a7     Document Type: Article

References (112)

1. A.B. Altman and S.L. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980) 50-112.
2. P.S. Aspinwall and D. Morrison, Topological field theory and rational curves, Commun. Math. Phys. 151 (1993) 245-262
3. V. Baranovsky, Moduli of sheaves on surfaces and action of the oscillator algebra
4. A. Beauville, Variétés kähleriennes dont la première classe de Chern est nulle, J. Diff. Geom. 18 (1983) 755-782.
5. A. Beauville, Counting rational curves on K3 surfaces, Duke Math. J. 97 (1999) 99-108
6. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997) 601-617
7. K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45-88
8. K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke. J. Math. 85 (1996) 1-60
9. A. Beilinson and A. Levin, The Elliptic polylogarithm, in : Motives, (U. Jannsen, S. Kleiman, J.-P. Serre, eds.), Proc. Symp. Pure Math. 55, Amer. Math. Soc., 1994, Part 2 123-190.
10. M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, (with an appendix by S. Katz), Nucl. Phys. B405 (1993) 279-304
11. 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) 311-427
12. M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Topological reduction of 4D SYM to 2D σ-models, Nucl. Phys. B448 (1995) 166
13. s+2,2(R{double-struck}) and infinite products, Invent. Math. 120 (1995) 161-213.
14. R.E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998) 491-562
15. S. Bradlow, Vortices in holomorphic line bundles over closed Kähler manifolds, Comm. Math. Phys. 135 (1990) 1-17.
16. S. Bradlow, G.D. Daskalopoulos, O. García-Prada, R. Wentworth, Stable augmented bundles over Riemann surfaces, in Vector bundles in algebraic geometry (Durham, 1993), (N.J. Hitchin, P.E. Newstead and W.M. Oxbury, eds), London Math. Soc. Lecture Note Ser., 208, Cambridge Univ. Press, 1995.
17. G.E. Bredon, Sheaf theory, 2nd edition, Graduate Texts in Mathematics 170, Springer-Verlag, 1997.
18. M. Camperi, F. Levstein and G. Zemba, The large N limit of Chern-Simons gauge theory, Phys. Lett. B247 (1990) 549-554.
19. J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. 109 (1979) 259-300.
20. J. Cheah, On the cohomology of Hilbert schemes of points, J. Alg. Geom. 5 (1996) 479-511.
21. N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, 1997.
22. H.R. Christiansen, C. Núñez and F.A. Schaposnik, Uniqueness of Bogomol'nyi equations and Born-Infeld like supersymmetric theories, Phys. Lett. B441 (1998) 185-190
23. P. Deligne, Théorie de Hodge III, Inst. Hautes Etudes Sci. Publ. Math. 44 (1974) 5-77.
24. R. Dijkgraaf, Mirror symmetry and elliptic curves, in The moduli space of curves, (R. Dijk-graaf, C. Faber and G. van der Geer, eds.), Progr. Math. 129, Birkhäuser, 1995.
25. R. Dijkgraaf, G. Moore, E. Verlinde and H. Verlinde, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys. 185 (1997) 197-209.
26. L.J. Dixon, V. Kaplunovsky and J. Louis, Moduli dependence of string loop corrections to gauge coupling constants, Nucl. Phys. B355 (1991) 649-688.
27. J. Edelstein, C. Núñez and F.A. Schaposnik, Supersymmetry and Bogomol'nyi equations in the Abelian Higgs Model, Phys. Lett. B329 (1994) 39-45.
28. M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Mathematics 55, Birkhäuser, 1985.
29. G. Ellingsrud, L. Götsche, M. Lehn, On the cobordism class of the Hilbert scheme of a surface
30. A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions, Vol.1, McGraw-Hill, 1953.
31. C. Faber, A conjectural description of the tautological ring of the moduli space of curves, preprint 1996.
32. C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory
33. B. Fantechi, L. Göttsche, D. van Straten, Euler number of the compactified Jacobian and multiplicity of rational curves, J. Alg. Geom. 8 (1999) 115-133
34. R. Fintushel and R.J. Stern, Knots, links, and 4-manifolds, Invent. Math. 134 (1998) 363-400
35. W. Fulton, Intersection Theory, 2nd edition, Springer-Verlag, 1998.
36. R. Friedman, Algebraic surfaces and holomorphic vector bundles, Springer-Verlag, 1998.
37. O. García-Prada, Seiberg-Witten invariants and vortex equations, in Symétries quantiques (A. Connes, K. Gawedzki and J. Zinn-Justin, eds), North-Holland, 1998.
38. E. Getzler and R. Pandharipande, Virasoro constraints and the Chern classes of the Hodge bundle, Nucl. Phys. B530 (1998) 701-714
39. L. Göttsche, Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Mathematics 1572, Springer-Verlag, 1994.
40. L. Göttsche and W. Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann. 296 (1993) 235-245.
41. + = 1, Selecta Math. (N.S.) 4 (1998) 69-115
42. R. Gopakumar and C. Vafa, Topological gravity as large N topological gauge theory, Adv. Theor. Math. Phys. 2 (1998) 413-442
43. R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence
44. R. Gopakumar and C. Vafa, M-theory and topological strings-I, II
45. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999) 487-518
46. V.A. Gritsenko and V.V. Nikulin, Siegel automorphic form corrections of some Lorentzian Kac-Moody Lie algebras, Amer. J. Math. 119 (1997) 181-224
47. I. Grojnowski, Instantons and affine algebras I: the Hilbert scheme and vertex operators, Math. Res. Letters 3 (1996) 275-291.
48. A. Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki, 1960/61, no.221.
49. J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986) 457-485.
50. J.A. Harvey and G. Moore, Algebras, BPS states, and strings, Nucl. Phys. B463 (1996) 315-368
51. J.A. Harvey and G. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998) 489-519
52. M. He, Espaces de modules de systèmes cohérents, Internat. J. Math. 9 (1998) 545-598.
53. D. Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988) 639-660.
54. S. Hosono, M.-H. Saito and A. Takahashi, Holomorphic anomaly equation and BPS state counting of rational elliptic surface, Adv. Theor. Math. Phys. 3 (1999) 177-208
55. B. Hunt and R. Schimmrigk, K3-fibered Calabi-Yau threefolds I, the twist map
56. D. Huybrechts, Birational symplectic manifolds and their deformations, J. Diff. Geom. 45 (1997) 488-513
57. D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Vieweg, 1997.
58. M. Jacob (ed.), Dual Theory, North-Holland, 1974.
59. D.P. Jatkar and B. Peeters, String theory near a conifold singularity, Phys. Lett. B362 (1995) 73-77
60. V.G. Kac, Infinite dimensional Lie algebras, 3rd edition, Cambridge University Press, 1990.
61. V.G. Kac, Vertex algebras for beginners, 2nd edition, American Mathematical Society, 1998.
62. M. Kaneko and D. Zagier, A generalized theta function and quasimodular forms, in The moduli space of curves, (R. Dijkgraaf, C. Faber and G. van der Geer, eds.), Progr. Math. 129, Birkhäuser 1995.
63. S. Katz, A. Klemm and C. Vafa, M-theory, topological strings and spinning black holes
64. T. Kawai, N = 2 heterotic string threshold correction, K3 surface and generalized Kac-Moody superalgebra, Phys. Lett. B372 (1996) 59-64.
65. T. Kawai, K3 surfaces, Igusa cusp form and string theory, in Topological field theory, primitive forms and related topics, (M. Kashiwara, A. Matsuo, K. Saito and I. Satake, eds.), Progr. Math. 160, Birkhäuser 1998.
66. T. Kawai, String duality and enumeration of curves by Jacobi forms, in Integrable systems and algebraic geometry, (M.-H. Saito, Y. Shimizu and K. Ueno, eds.), World Scientific 1998
67. T. Kawai, Y. Yamada and S.-K. Yang, Elliptic genera and N=2 superconformal field theory, Nucl. Phys. B414 (1994) 191-212.
68. A. Klemm and E. Zaslow, Local mirror symmetry at higher genus
69. M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120-139, Birkhäuser, Basel, 1995.
70. M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumer-ative geometry, Commun. Math. Phys. 164 (1994) 525-562
71. M. Kontsevich and Yu. Manin, Relations between the correlators of the topological sigma-model coupled to gravity, Commun. Math. Phys. 196 (1988) 385-398
72. M. Lehn, Chern classes of tautological sheaves on Hilbert schemes
73. J. Le Potier, Systèmes cohérents et structures de niveau, Astérisque 214 (1993).
74. J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, Jour, AMS 11 (1998) 119-174
75. M. Lübke and A. Teleman, The Kobayashi-Hitchin correspondence, World Scientific, 1995.
76. I.G. Macdonald, Symmetric products of an algebraic curve, Topology, 1 (1962) 319-343.
77. Yu. Manin, Generating functions in algebraic geometry and sums over trees, in The moduli space of curves, (R. Dijkgraaf, C. Faber and G. van der Geer, eds.), Progr. Math. 129, Birkhäuser 1995
78. E. Markman, Brill-Noether duality for moduli spaces of sheaves on K3 surfaces
79. M. Mariño and G. Moore, Counting higher genus curves in a Calabi-Yau manifold, Nucl. Phys. B543 (1999) 592-614
80. M. Mariño and G. Moore, 3-manifold topology and the Donaldson-Witten partition function, Nucl. Phys. B547 (1999) 569-598
81. G. Meng and C. Taubes, SW=Milnor torsion, Math. Res. Lett. 3 (1996) 661-674.
82. G. Moore and E. Witten, Integration over the u-plane in Donaldson theory, Adv. Theor. Math. Phys. 1 (1997) 298-387
83. J.W. Morgan and Z. Szabó, Embedded tori in four-manifolds, Topology 38 (1999) 479-496.
84. D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds - II Nucl. Phys. B476 (1996) 437-469
85. W. Müller, Analytic torsion and R-torsion on Riemannian manifolds, Adv. in Math. 28 (1978) 233-305.
86. S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984) 101-106.
87. S. Mukai, On the moduli space of bundles on K3 surfaces I, in Vector bundles on algebraic varieties, Tata Inst. Fund. Res, 1987.
88. S. Mukai, Moduli of vector bundles on K3 surfaces and symplectic manifolds, Sugaku Expositions 1 (1988) 139-174.
89. D. Mumford, Towards an enumerative geometry of the moduli space of curves, in Arithmetic and geometry, Vol. II, 271-328, Progr. Math. 36, Birkhäuser, 1983.
90. H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. Math. 145 (1997) 379-388.
91. H. Nakajima, Lectures on Hilbert schemes of points on surfaces, American Mathematical Society, 1999.
92. K. O'Grady, The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface, J. Alg. Geom. 6 (1997) 599-644.
93. K. O'Grady, Desingularized moduli spaces of sheaves on a K3, J. reine angew. Math. 512 (1999) 49-117.
94. R. Pandharipande, Hodge integrals and degenerate contributions, Commun. Math. Phys. 208 (1999) 489-506
95. V. Periwal, Topological closed-string interpretation of Chern-Simons theory, Phys. Rev. Lett. 71 (1993) 1295-1298
96. D. Ray and I. Singer, R-torsion and the Laplacian in Riemannian manifolds, Adv. in Math. 7 (1971) 145-210.
97. C. Truesdell, On a function which occurs in the theory of the structure of polymers, Ann. of Math. 46 (1945) 144-157.
98. V. Turaev, Reidemeister torsion in knot theory, Russ. Math. Surv. 41 (1986) 119-182.
99. C. Vafa and E. Witten, A strong coupling test for S-duality, Nucl. Phys. B431 (1994) 3-77
100. A. Weil, Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Mathe-matik und ihrer Grenzgebiete, Band 88, Springer-Verlag, 1976.
101. K. Wirthmüller, Root systems and Jacobi forms, Compositio Math. 82 (1992) 293-354.
102. E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351.
103. E. Witten, On quantum gauge theories in two dimensions, Commun. Math. Phys. 141 (1991) 153-209.
104. E. Witten, Mirror manifolds and topological field theory, in Essays on mirror manifolds, (S.-T. Yau ed.), Internat. Press, 1992
105. E. Witten, Chern-Simons gauge theory as a string theory, in The Floer memorial volume, (H. Hofer, C.H. Taubes, A. Weinstein and E. Zehnder eds.), Progr. Math. 133, Birkhäuser 1995
106. E. Witten, Phases of N = 2 theories in two dimensions, Nucl. Phys. B403 (1993) 159-222
107. E. Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994) 769-796
108. S.-T. Yau and E. Zaslow, BPS states, string duality, and nodal curves on K3, Nucl. Phys. B471 (1996) 503-512
109. K. Yoshioka, Some examples of Mukai's reflections on K3 surfaces, J. reine angew. Math. 515 (1999) 97-123.
110. K. Yoshioka, Irreducibility of moduli spaces of vector bundles on K3 surfaces
111. D. Zagier, The Bloch-Wigner-Ramakrishnan polylogarithm function, Math. Ann. 286 (1990) 613-624.
112. D. Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math. 104 (1991) 449-465.