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Letters in Mathematical Physics

, Volume 74, Issue 1, pp 53–74 | Cite as

Khovanov-Rozansky Homology and Topological Strings

  • Sergei GukovEmail author
  • Albert Schwarz
  • Cumrun Vafa
Article
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Abstract

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological string theory. We use this approach in various examples to predict the sl(N) knot homology groups for all values of N. We verify that our predictions pass some non-trivial checks

Keywords

knots quantum group invariants knot homology topological strings BPS states 
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References

  1. 1.
    Schwarz A. New topological invariants arising in the theory of quantized fields. Baku International Topological Conference Abstracts (part II) (1987)Google Scholar
  2. 2.
    Witten E. (1989). Quantum field theory and the jones polynomial. Commun. Math. Phys. 121:351zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Freyd P., Yetter D., Hoste J., Lickorish W., Millett K., Oceanu A. (1985). A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. 12:239zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Witten E. (1995). Chern-Simons gauge theory as a string theory. Prog. Math. 133:637 hep-th/9207094MathSciNetGoogle Scholar
  5. 5.
    Gopakumar R., Vafa C. (1999). On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3:1415 hep-th/9811131zbMATHMathSciNetGoogle Scholar
  6. 6.
    Gopakumar, R., Vafa, C.: M-theory and topological strings. I,II. hep-th/9809187; hep-th/9812127Google Scholar
  7. 7.
    Ooguri H., Vafa C. (2000). Knot invariants and topological strings. Nucl. Phys. B577:419zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Labastida J.M.F., Marino M., Vafa C. (2000). Knots, links and branes at large N. JHEP 0011:007 hep-th/0010102CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Hosono S., Saito M.-H., Takahashi A. (1999). Holomorphic anomaly equation and BPS state counting of rational elliptic surface. Adv. Theor. Math. Phys. 3:177zbMATHMathSciNetGoogle Scholar
  10. 10.
    Hosono S., Saito M.-H., Takahashi A. (2001). Relative Lefschetz action and BPS state counting. Int. Math. Res. Notices 15:783CrossRefMathSciNetGoogle Scholar
  11. 11.
    Katz S., Klemm A., Vafa C. (1999). M-theory, topological strings and spinning black holes. Adv. Theor. Math. Phys. 3:1445 hep-th/9910181zbMATHMathSciNetGoogle Scholar
  12. 12.
    Maulik D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and Donaldson-Thomas theory, I. math.AG/0312059Google Scholar
  13. 13.
    Katz, S.: Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds. math.ag/0408266Google Scholar
  14. 14.
    Khovanov M. (2000). A categorification of the Jones polynomial. Duke Math. J. 101:359 math.QA/9908171zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Khovanov M. (2005). Categorifications of the colored Jones polynomial. J. Knot Theory Ramifications 14:111 math.QA/0302060zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Khovanov M. (2004). sl(3) link homology. Algebr. Geom. Topol. 4:1045 math.QA/0304375zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Khovanov, M., Rozansky, L.: Matrix factorizations and link homology. math.QA/ 0401268Google Scholar
  18. 18.
    Bar-Natan D. (2002). On Khovanov’s categorification of the Jones polynomial. Algebr. Geom. Topol. 2:337 math.QA/0201043zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Khovanov, M.: An invariant of tangle cobordisms. math.QA/0207264Google Scholar
  20. 20.
    Jacobsson M. (2004). An invariant of link cobordisms from Khovanov’s homology. Algebr. Geom. Topol. 4:1211 math.GT/0206303zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Shumakovitch, A.: KhoHo – a program for computing and studying Khovanov homology. http://www.geometrie.ch/KhoHoGoogle Scholar
  22. 22.
    Hollowood, T.J., Iqbal, A., Vafa, C.: Matrix models, geometric engineering and elliptic genera. hep-th/0310272Google Scholar
  23. 23.
    Labastida J.M.F., Marino M. (2001). Polynomial invariants for torus knots and topological strings. Commun. Math. Phys. 217:423 hep-th/0004196zbMATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Labastida, J.M.F., Marino, M.: A new point of view in the theory of knot and link invariants. math.qa/0104180Google Scholar
  25. 25.
    Murakami H., Ohtsuki T., Yamada S. (1998). HOMFLY polynomial via an invariant of colored plane graphs. Enseign. Math. 44:325zbMATHMathSciNetGoogle Scholar
  26. 26.
    Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. hep-th/0403166Google Scholar
  27. 27.
    Kontsevich, M.: unpublishedGoogle Scholar
  28. 28.
    Kapustin A., Li Y. (2003). D-branes in Landau-Ginzburg models and algebraic geometry. JHEP 0312:005 hep-th/0210296CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Orlov D. (2004). Triangulated categories of singularities and D-branes in Landau-Ginzburg orbifold. Proc. Steklov Inst. Math. 246:227 math.AG/0302304MathSciNetGoogle Scholar
  30. 30.
    Brunner, I., Herbst, M., Lerche, W., Scheuner, B.: Landau-Ginzburg realization of open string TFT. hep-th/0305133Google Scholar
  31. 31.
    Ashok, S.K., Dell’Aquila, E., Diaconescu, D.E.: Fractional branes in Landau-Ginzburg orbifolds. hep-th/0401135Google Scholar
  32. 32.
    Ashok, S.K., Dell’Aquila, E., Diaconescu, D.E., Florea, B.: Obstructed D-branes in Landau-Ginzburg orbifolds. hep-th/0404167Google Scholar
  33. 33.
    Hori K., Walcher J. (2005). F-term equations near Gepner points. JHEP 0501:008 hep-th/0404196CrossRefADSMathSciNetGoogle Scholar
  34. 34.
    Brunner, I., Herbst, M., Lerche, W., Walcher, J.: Matrix factorizations and mirror symmetry: the cubic curve. hep-th/0408243Google Scholar
  35. 35.
    Mirror symmetry (Clay mathematics monographs, V. 1). In: Hori, K. et al. (ed.) American Mathematical Society (2003)Google Scholar
  36. 36.
    Schwarz, A., Shapiro, I.: Some remarks on Gopakumar–Vafa invariants. hep-th/0412119Google Scholar
  37. 37.
    Taubes C. (2001). Lagrangians for the Gopakumar–Vafa conjecture. Adv. Theor. Math. Phys. 5:139 math.DG/0201219zbMATHMathSciNetGoogle Scholar
  38. 38.
    Dunfield, N., Gukov, S., Rasmussen, J.: The superpolynomial for knot homologies. math.GT/0505662Google Scholar
  39. 39.
    Bar-Natan, D.: Some Khovanov-Rozansky Computations. http://www.math.toronto.edu/ drorbn/Misc/KhovanovRozansky/index.htmlGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Jefferson Physical LaboratoryHarvard UniversityCambridgeUSA
  2. 2.Department of MathematicsUniversity of CaliforniaDavisUSA

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