Letters in Mathematical Physics

, Volume 103, Issue 12, pp 1329–1345 | Cite as

Uniqueness Theorem of \({\mathcal{W}}\) -Constraints for Simple Singularities

  • Si-Qi Liu
  • Di Yang
  • Youjin ZhangEmail author


In a recent paper, Bakalov and Milanov (Compositio. Math. 149: 840–888, 2013) proved that the total descendant potential of a simple singularity satisfies the \({\mathcal{W}}\) -constraints, which come from the \({\mathcal{W}}\) -algebra of the lattice vertex algebra associated with the root lattice of this singularity and a twisted module of the vertex algebra. In the present paper, we prove that the solution of these \({\mathcal{W}}\) -constraints is unique up to a constant factor, as conjectured by Bakalov and Milanov in their paper.


W-constraint simple singularity vertex algebra 

Mathematics Subject Classification (2010)

Primary 53D45 Secondary 17B69 32S30 81R10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler M., Moerbeke P.: A matrix integral solution to two-dimensional Wp -gravity. Commun. Math. Phys. 147, 25–56 (1992)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Bakalov B., Kac V.: Twisted modules over lattice vertex algebras. Lie Theory and Its Applications in Physics V, pp. 3–26. World Scientific, River Edge (2004)Google Scholar
  3. 3.
    Bakalov, B., Milanov, T.: \({\mathcal{W}}\) -constraints for the total descendant potential of a simple singularity. Compositio. Math. 149, 840-888 (2013)Google Scholar
  4. 4.
    Behrend K., Fantechi B.: The intrinsic normal cone. Invent. Math. 128, 45–88 (1997)MathSciNetADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Chiodo A.: The Witten top Chern class via K-theory. J. Algebraic Geom. 15, 681–707 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dijkgraaf R., Verlinde H., Verlinde E.: Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity. Nucl. Phys. B 348, 435–456 (1991)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups (Montecatini Terme, 1993). Lecture Notes in Mathematics, vol. 1620, pp. 120–348. Springer, Berlin (1996)Google Scholar
  8. 8.
    Dubrovin B.: Painlevé transcendents in two-dimensional topological field theory. The Painlevé property. CRM Series in Mathematical Physics, pp. 287–412. Springer, New York (1999)Google Scholar
  9. 9.
    Dubrovin, B., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. eprint arXiv: math/0108160Google Scholar
  10. 10.
    Dubrovin B., Zhang Y.: Virasoro symmetries of the extended Toda hierarchy. Commun. Math. Phys. 250, 161–193 (2004)MathSciNetADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Eguchi T., Yang S.-K.: The topological \({\mathbb{CP}^1}\) model and the large-N matrix integral. Mod. Phys. Lett. A 9, 2893–2902 (1994)MathSciNetADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Faber C., Shadrin S., Zvonkine D.: Tautological relations and the r-spin Witten conjecture. Ann. Sci. éc. Norm. Supér 43, 621–658 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fateev V.A., Lukyanov S.L.: The models of two-dimensional conformal quantum field theory with \({\mathbb{Z}}\) n symmetry. Int. J. Mod. Phys. A 3, 507–520 (1988)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Fan, H., Jarvis, T.J., Merrell, E., Ruan, Y.: Witten’s D 4 Integrable Hierarchies Conjecture. eprint arXiv: 1008.0927Google Scholar
  15. 15.
    Fan, H., Jarvis, T.J., Ruan, Y.: The Witten equation, mirror symmetry and quantum singularity theory. Ann. Math. 178, 1–106 (2013)Google Scholar
  16. 16.
    Fan, H., Jarvis, T.J., Ruan, Y.: The Witten equation and its virtual fundamental cycle. eprint arXiv: 0712.4025Google Scholar
  17. 17.
    Feigin B., Frenkel E.: Quantization of the Drinfeld–Sokolov reduction. Phys. Lett. B 246, 75–81 (1990)MathSciNetADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Feigin, B., Frenkel, E.: Integrals of motion and quantum groups. Integrable systems and quantum groups (Montecatini Terme, 1993). Lecture Notes in Mathematics, vol. 1620, pp. 349–418. Springer, Berlin (1996)Google Scholar
  19. 19.
    Frenkel E., Givental A., Milanov T.: Soliton equations, vertex operators, and simple singularities. Funct. Anal. Other Math. 3, 47–63 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Frenkel, I.B.: Representations of affine Lie algebras, Hecke modular forms and Korteweg-de Vries type equations. Lie algebras and related topics (New Brunswick, N.J., 1981). Lecture Notes in Mathematics, vol. 933, pp. 71–110. Springer, Berlin (1982)Google Scholar
  21. 21.
    Fukuma M., Kawai H., Nakayama R.: Continuum Schwinger–Dyson equations and universal structures in two-dimensional quantum gravity. Int. J. Mod. Phys. A 6, 1385–1406 (1991)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Fukuma M., Kawai H., Nakayama R.: Infinite dimensional Grassmannian structure of two-dimensional quantum gravity. Commun. Math. Phys. 143, 371–403 (1992)MathSciNetADSCrossRefzbMATHGoogle Scholar
  23. 23.
    Getzler E.: The Toda conjecture. Symplectic geometry and mirror symmetry (Seoul, 2000), pp. 51–79. World Scientific, River Edge (2001)CrossRefGoogle Scholar
  24. 24.
    Givental, A.B.: Gromov-Witten invariants and quantization of quadratic Hamiltonians. Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. Mosc. Math. J. 1, 551–568, 645 (2001)Google Scholar
  25. 25.
    Goeree J.: \({\mathcal{W}}\) -constraints in 2D quantum gravity. Nucl. Phys. B 358, 737–757 (1991)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Humphreys J.E.: Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, 29. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  27. 27.
    Kac V.: Vertex algebras for beginners, 2nd edn. University Lecture Series 10. American Mathematical Society, Providence (1998)Google Scholar
  28. 28.
    Kac V., Schwarz A.: Geometric interpretation of the partition function of 2D gravity. Phys. Lett. B 257, 329–334 (1991)MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy functionx. Commun. Math. Phys. 147, 1–23 (1992)MathSciNetADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Kontsevich M., Manin Yu.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994)MathSciNetADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Milanov T.: Hirota quadratic equations for the extended Toda hierarchy. Duke Math. J. 138, 161–178 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Okounkov A., Pandharipande R.: The equivariant Gromov-Witten theory of \({\mathbb{P}^1}\) . Ann. Math. 163, 561–605 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Polishchuk, A., Vaintrob, A.: Algebraic construction of Witten’s top Chern class. Advances in algebraic geometry motivated by physics (Lowell, MA, 2000). Contemporary Mathematics, vol. 276, pp. 229–249. American Mathematical Society, Providence (2001)Google Scholar
  34. 34.
    Ruan Y., Tian G.: A mathematical theory of quantum cohomology. J. Differ. Geom. 42, 259–367 (1995)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Teleman C.: The structure of 2D semi-simple field theories. Invent. Math. 188, 525–588 (2012)MathSciNetADSCrossRefzbMATHGoogle Scholar
  36. 36.
    Witten, E.: Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990), pp. 243–310. Lehigh University, Bethlehem (1991)Google Scholar
  37. 37.
    Witten, E.: Algebraic geometry associated with matrix models of two-dimensional gravity. Topological methods in modern mathematics (Stony Brook, NY, 1991), pp. 235–269. Publish or Perish, Houston (1993)Google Scholar
  38. 38.
    Wu C.-Z.: A remark on Kac–Wakimoto hierarchies of D-type. J. Phys. A 43, 035201 (2010)MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    Zamolodchikov A.B.: Infinite extra symmetries in two-dimensional conformal quantum field theory. (Russian) Teoret. Mat. Fiz. 65, 347–359 (1985)MathSciNetGoogle Scholar
  40. 40.
    Zhang Y.: On the \({\mathbb{CP}^{1}}\) topological sigma model and the Toda lattice hierarchy. J. Geom. Phys. 40, 215–232 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina

Personalised recommendations