Geometry as seen by string theory



This is an introductory review of the topological string theory from physicist’s perspective. I start with the definition of the theory and describe its relation to the Gromov–Witten invariants. The BCOV holomorphic anomaly equations, which generalize the Quillen anomaly formula, can be used to compute higher genus partition functions of the theory. The open/closed string duality relates the closed topological string theory to the Chern–Simons gauge theory and the random matrix model. As an application of the topological string theory, I discuss the counting of bound states of D-branes.

Keywords and phrases:

topological string theory 

Mathematics Subject Classification (2000):

14N35 81T30 


  1. 1.
    M. Aganagic, A. Klemm, M. Marino and C. Vafa, The topological vertex, Comm. Math. Phys., 254 (2005), 425–478, arXiv:hep-th/0305132.Google Scholar
  2. 2.
    M. Aganagic, H. Ooguri, N. Saulina and C. Vafa, Black holes, q-deformed 2d Yang–Mills, and non-perturbative topological strings, Nuclear Phys. B, 715 (2005), 304–348, arXiv:hepth/ 0411280.Google Scholar
  3. 3.
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rep., 323 (2000), 183–386, arXiv:hep-th/9905111.Google Scholar
  4. 4.
    P.S. Aspinwall, B.R. Greene and D.R. Morrison, Multiple mirror manifolds and topology change in string theory, Phys. Lett. B, 303 (1993), 249–259, arXiv:hep-th/9301043; Calabi–Yau moduli space, mirror manifolds and spacetime topology change in string theory, Nuclear Phys. B, 416 (1994), 414–480, arXiv:hep-th/9309097.Google Scholar
  5. 5.
    P.S. Aspinwall and D.R. Morrison, Topological field theory and rational curves, Comm. Math. Phys., 151 (1993), 245–262, arXiv:hep-th/9110048.Google Scholar
  6. 6.
    C. Beasley, D. Gaiotto, M. Guica, L. Huang, A. Strominger and X. Yin, Why Z BH = |Z top|2, arXiv:hep-th/0608021.Google Scholar
  7. 7.
    K. Becker, M. Becker and A. Strominger, Fivebranes, membranes and non-perturbative string theory, Nuclear Phys. B, 456 (1995), 130–152, arXiv:hep-th/9507158.Google Scholar
  8. 8.
    J.D. Bekenstein, Black holes and entropy, Phys. Rev. D (3), 7 (1973), 2333–2346.CrossRefMathSciNetGoogle Scholar
  9. 9.
    M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nuclear Phys. B, 405 (1993), 279–304, arXiv:hep-th/9302103.Google Scholar
  10. 10.
    M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys., 165 (1994), 311–427, arXiv:hep-th/9309140.Google Scholar
  11. 11.
    J. de Boer, M.C.N. Cheng, R. Dijkgraaf, J. Manschot and E. Verlinde, A Farey tail for attractor black holes, J. High Energy Phys., 2006 (2006), 024, pp. 28, arXiv:hep-th/0608059.Google Scholar
  12. 12.
    P. Candelas, X.C. de la Ossa, P.S. Green, and L. Parkes, A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B, 359 (1991) :21–74.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    G.L. Cardoso, B. de Wit and T. Mohaupt, Corrections to macroscopic supersymmetric black-hole entropy, Phys. Lett. B, 451 (1999), 309–316, arXiv:hep-th/9812082.Google Scholar
  14. 14.
    S. Cecotti and C. Vafa, Topological–anti-topological fusion, Nuclear Phys. B, 367 (1991), 359–461.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Precision counting of small black holes, J. High Energy Phys., 2005 (2005), 096, pp. 90, arXiv:hep-th/0507014.Google Scholar
  16. 16.
    F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, arXiv:hepth/0702146.Google Scholar
  17. 17.
    R. Dijkgraaf and C. Vafa, Matrix models, topological strings, and supersymmetric gauge theories, Nuclear Phys. B, 644 (2002), 3–20, arXiv:hep-th/0206255; On geometry and matrix models, Nuclear Phys. B, 644 (2002), 21–39, arXiv:hep-th/0207106.Google Scholar
  18. 18.
    R. Dijkgraaf, C. Vafa and E. Verlinde, M-theory and a topological string duality, arXiv:hepth/ 0602087.Google Scholar
  19. 19.
    J. Distler and B.R. Greene, Some exact results on the superpotential from Calabi–Yau compactifications, Nuclear Phys. B, 309 (1988), 295–316.CrossRefMathSciNetGoogle Scholar
  20. 20.
    S.K. Donaldson and R.P. Thomas, Gauge theory in higher dimensions, In: The Geometric Universe, Oxford Univ. Press, 1998, pp. 31–47.Google Scholar
  21. 21.
    C. Faber and R. Pandharipande, Hodge integrals and Gromov–Witten theory, arXiv: math/9810173.Google Scholar
  22. 22.
    H. Fang, Z. Lu and K.-I. Yoshikawa, Analytic torsion for Calabi–Yau threefolds, arXiv: math/0601411.Google Scholar
  23. 23.
    D. Gaiotto, A. Strominger and X. Yin, From AdS3/CFT2 to black holes/topological strings, J. High Energy Phys., 2007 (2007), 050, pp. 10, arXiv:hep-th/0602046.Google Scholar
  24. 24.
    R. Gopakumar and C. Vafa, Topological gravity as large N topological gauge theory, Adv. Theor. Math. Phys., 2 (1998), 413–442, arXiv:hep-th/9802016.Google Scholar
  25. 25.
    R. Gopakumar and C. Vafa, M-theory and topological strings. I, arXiv:hep-th/9809187.Google Scholar
  26. 26.
    R. Gopakumar and C. Vafa, M-theory and topological strings. II, arXiv:hep-th/9812127.Google Scholar
  27. 27.
    B.R. Greene and M.R. Plesser, Duality in Calabi–Yau moduli space, Nuclear Phys. B, 338 (1990), 15–37.CrossRefMathSciNetGoogle Scholar
  28. 28.
    R.K. Gupta and A. Sen, AdS3/CFT2 to AdS2/CFT1, arXiv:0806.0053.Google Scholar
  29. 29.
    S.W. Hawking, Gravitational radiation from colliding black holes, Phys. Rev. Lett., 26 (1971), 1344–1346.Google Scholar
  30. 30.
    S.W. Hawking, Black hole explosions?, Nature, 248 (1974), p. 30.Google Scholar
  31. 31.
    W. Heisenberg and W. Pauli, Zur Quantendynamik derWellenfelder, Z. Phys. A, 56 (1929), 1–61.Google Scholar
  32. 32.
    G. ’t Hooft, A planar diagram theory for strong interactions, Nuclear Phys. B, 72 (1974), 461–473.Google Scholar
  33. 33.
    M.-X. Huang, A. Klemm and S. Quackenbush, Topological string theory on compact Calabi–Yau: modularity and boundary conditions, arXiv:hep-th/0612125.Google Scholar
  34. 34.
    A. Iqbal, C. Vafa, N. Nekrasov and A. Okounkov, Quantum foam and topological strings, J. High Energy Phys., 2008 (2008), 011, pp. 47, arXiv:hep-th/0312022.Google Scholar
  35. 35.
    K. Liu and P. Peng, Proof of the Labastida–Marino–Ooguri–Vafa conjecture, arXiv:0704.1526.Google Scholar
  36. 36.
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys., 2 (1998), 231–252, Internat. J. Theoret. Phys., 38 (1999), 1113–1133, arXiv:hep-th/9711200.Google Scholar
  37. 37.
    M. Marino, Chern–Simons Theory, Matrix Models, and Topological Strings, Internat. Ser. Monogr. Phys., 131, Oxford Univ. Press, 2005.Google Scholar
  38. 38.
    D. Maulik, A. Oblomkov, A. Okounkov and R. Pandharipande, Gromov–Witten/Donaldson–Thomas correspondence for toric 3-folds, arXiv:0809.3976.Google Scholar
  39. 39.
    S. Mozgovoy and M. Reineke, On the noncommutative Donaldson–Thomas invariants arising from brane tilings, arXiv:0809.0117.Google Scholar
  40. 40.
    H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras, Duke Math. J., 76 (1994), 365–416.Google Scholar
  41. 41.
    N. Nekrasov and A. Okounkov, Seiberg–Witten theory and random partitions, arXiv:hep-th/ 0306238.Google Scholar
  42. 42.
    A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi–Yau and classical crystals, arXiv:hep-th/0309208.Google Scholar
  43. 43.
    H. Ooguri, Y. Oz and Z. Yin, D-branes on Calabi–Yau spaces and their mirrors, Nuclear Phys. B, 477 (1996), 407–430, arXiv:hep-th/9606112.Google Scholar
  44. 44.
    H. Ooguri, A. Strominger and C. Vafa, Black hole attractors and the topological string, Phys. Rev. D (3), 70 (2004), 106007, pp. 13, arXiv:hep-th/0405146.Google Scholar
  45. 45.
    H. Ooguri and C. Vafa, Knot invariants and topological strings, Nuclear Phys. B, 577 (2000), 419–438, arXiv:hep-th/9912123.Google Scholar
  46. 46.
    H. Ooguri and C. Vafa, Worldsheet derivation of a large N duality, Nuclear Phys. B, 641 (2002), 3–34, arXiv:hep-th/0205297.Google Scholar
  47. 47.
    H. Ooguri and M. Yamazaki, Crystal melting and toric Calabi–Yau manifolds, arXiv:0811.2801.Google Scholar
  48. 48.
    J. Polchinski, Dirichlet-branes and Ramond–Ramond charges, Phys. Rev. Lett., 75 (1995), 4724–4727, arXiv:hep-th/9510017.Google Scholar
  49. 49.
    D.B. Ray and I.M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2), 98(1973), 154–177.Google Scholar
  50. 50.
    G. Segal, The definition of conformal field theory, In: Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., First circulated in 1988, pp. 421–577.Google Scholar
  51. 51.
    G. Segal, Two-dimensional conformal field theories and modular functors, In: IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, 1989, pp. 22–37.Google Scholar
  52. 52.
    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein–Hawking entropy, Phys. Lett. B, 379 (1996), 99–104, arXiv:hep-th/9601029.Google Scholar
  53. 53.
    B. Szendröi, Non-commutative Donaldson–Thomas invariants and the conifold, Geom. Topol., 12 (2008), 1171–1202, arXiv:0705.3419.Google Scholar
  54. 54.
    T. Takagi, Über eine Theorie des relativ Abel’schen Zahlkörpers, J. Coll. Sci. Imp. Univ. Tokyo, 41 (1920), 1–133.Google Scholar
  55. 55.
    T. Takagi, Kaiseki Gairon, Iwanami Shoten, 1983.Google Scholar
  56. 56.
    T. Takagi, Kinsei Suugaku Shidan, Iwanami Shoten, 1995.Google Scholar
  57. 57.
    C. Vafa, Two dimensional Yang–Mills, black holes and topological strings, arXiv:hepth/0406058.Google Scholar
  58. 58.
    C. Vafa and E. Witten, A strong coupling test of S-duality, Nuclear Phys. B, 431 (1994), 3–77, arXiv:hep-th/9408074.Google Scholar
  59. 59.
    E. Witten, Noncommutative geometry and string field theory, Nuclear Phys. B, 268 (1986), 253–294.Google Scholar
  60. 60.
    E. Witten, Topological sigma models, Comm. Math. Phys., 118 (1988), 411–449.Google Scholar
  61. 61.
    E.Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys., 121 (1989), 351–399.Google Scholar
  62. 62.
    E. Witten, Phases of \({\mathcal{N}} = 2\) theories in two dimensions, Nuclear Phys. B, 403 (1993), 159–222, arXiv:hep-th/9301042.Google Scholar
  63. 63.
    E. Witten, Chern–Simons gauge theory as a string theory, Prog. Math., 133 (1995), 637– 678, arXiv:hep-th/9207094.Google Scholar
  64. 64.
    E. Witten, Mirror manifolds and topological field theory, In: Mirror Symmetry I, (ed. S.-T. Yau), Amer. Math. Soc., 1998, pp. 121–160, arXiv:hep-th/9112056.Google Scholar
  65. 65.
    S. Yamaguchi and S.T. Yau, Topological string partition functions as polynomials, J. High Energy Phys., 2004 (2004), 047, pp. 20, arXiv:hep-th/0406078.Google Scholar
  66. 66.
    A. Zinger, The reduced genus-one Gromov–Witten invariants of Calabi–Yau hypersurfaces, arXiv:0705.2397.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2009

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan

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