# BCFG Drinfeld–Sokolov hierarchies and FJRW-theory

Article

First Online:

Received:

Accepted:

- 868 Downloads
- 3 Citations

## Introduction

In 1991, Witten [
41] proposed a remarkable conjecture relating the intersection theory of the Deligne–Mumford moduli space
\({\overline{\mathcal {M}}}_{g,k}\)

## Notes

### Acknowledgments

The first and third authors would like to thank Boris Dubrovin for his encouragements and helpful discussions. Part of their work was supported by NSFC No. 11071135, No. 11171176, No. 11222108, and No. 11371214, and by the Marie Curie IRSES project RIMMP. The second author would like to thank his collaborator Todor Milanov from whom he learned most of his knowledge of integrable hierarchies. A special thanks goes to Edward Witten for inspiration and guidance on integrable hierarchies mirror symmetry. His work is partly supported by NSF grant DMS-1103368 and NSF FRG grant DMS-1159265.

## References

- 1.Balog, J., Fehér, L., O’Raifeartaigh, L., Forgács, P., Wipf, A.: Toda theory and \(W\)-algebra from a gauged WZNW point of view. Ann. Phys.
**203**(1990), 76–136 (1990)CrossRefGoogle Scholar - 2.Buryak, A., Posthuma, H., Shadrin, S.: On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket. J. Geom. Phys.
**62**, 1639–1651 (2012)MathSciNetCrossRefGoogle Scholar - 3.Buryak, A., Posthuma, H., Shadrin, S.: A polynomial bracket for the Dubrovin-Zhang hierarchies. J. Diff. Geom.
**92**, 153–185 (2012)MathSciNetGoogle Scholar - 4.Chiodo, A., Iritani, H., Ruan, Y.: Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence. Publ. Math. Inst. Hautes tudes Sci.
**119**, 127–216 (2014)MathSciNetCrossRefGoogle Scholar - 5.Drinfeld, V., Sokolov, V.: Lie algebras and equations of Korteweg-de Vries type. J. Soviet Math.
**30**, 1975–2036 (1985). Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya)**24**, 81–180 (1984)Google Scholar - 6.Dubrovin, B.: Integrable systems and classification of 2-dimensional topological field theories. In: Integrable Systems (Luminy, 1991), pp. 313–359. Progress in Mathematics, vol. 115, Birkhäuser, Boston (1993)Google Scholar
- 7.Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable systems and Quantum Groups (Montecatini Terme, 1993), pp. 120–348. Lecture Notes in Mathematics, vol. 1620, Springer, Berlin (1996)Google Scholar
- 8.Dubrovin, B., Liu, S.-Q., Zhang, Y.: On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasi-triviality of bi-Hamiltonian perturbations. Comm. Pure Appl. Math.
**59**, 559–615 (2006)Google Scholar - 9.Dubrovin, B., Liu, S.-Q., Zhang, Y.: Frobenius manifolds and central invariants for the Drinfeld-Sokolov bihamiltonian structures. Adv. Math.
**219**, 780–837 (2008)MathSciNetCrossRefGoogle Scholar - 10.Dubrovin, B., Zhang, Y.: Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation. Comm. Math. Phys.
**198**, 311–361 (1998)MathSciNetCrossRefGoogle Scholar - 11.Dubrovin, B., Zhang, Y.: Frobenius manifolds and Virasoro constraints. Selecta Math. (N.S.)
**5**, 423–466 (1999)Google Scholar - 12.Dubrovin, B., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. eprint arXiv:math/0108160
- 13.Dubrovin, B., Zhang, Y.: Virasoro Symmetries of the Extended Toda Hierarchy. Comm. Math. Phys.
**250**, 161–193 (2004)MathSciNetCrossRefGoogle Scholar - 14.Faber, C., Shadrin, S., Zvonkine, D.: Tautological relations and the r-spin Witten conjecture. Ann. Sci. Éc. Norm. Supér.
**43**, 621–658 (2010)Google Scholar - 15.Fan, H., Jarvis, T., Ruan, Y.: The Witten equation and its virtual fundamental cycle. eprint arXiv:math/0712.4025
- 16.Fan, H., Jarvis, T., Ruan, Y.: The Witten equation, mirror symmetry and quantum singularity theory. Ann. Math.
**178**, 1–106 (2013)Google Scholar - 17.Fan, H., Francis, A., Jarvis, T., Merrell, E., Ruan, Y.: Witten’s \(D_4\) Integrable hierarchies conjecture. eprint arXiv:1008.0927
- 18.Frenkel, E., Givental, A., Milanov, T.: Soliton equations, vertex operators, and simple singularities. Funct. Anal. Other Math.
**3**, 47–63 (2010)MathSciNetCrossRefGoogle Scholar - 19.Getzler, E.: The Toda conjecture. In: Symplectic Geometry and Mirror Symmetry (Seoul, 2000), pp. 51–79. World Scientific Publishing, River Edge (2001)Google Scholar
- 20.Givental, A.: Semisimple Frobenius structures at higher genus. Internat. Math. Res. Notices
**2001**(23), 1265–1286 (2001)MathSciNetCrossRefGoogle Scholar - 21.Givental, A.: 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 - 22.Givental, A., Milanov, T.: Simple singularities and integrable hierarchies. In: The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol. 232, pp. 173–201. Birkhauser Boston, Boston (2005)Google Scholar
- 23.Hollowood, T., Miramontes, J.: Tau-functions and generalized integrable hierarchies. Comm. Math. Phys.
**157**, 99–117 (1993)MathSciNetCrossRefGoogle Scholar - 24.Johnson, P.: Equivariant Gromov-Witten theory of one dimensional stacks. eprint arXiv:0903.1068
- 25.Kac, V.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
- 26.Kac, V., Wakimoto, M.: Exceptional hierarchies of soliton equations. In: Theta Functions-Bowdoin 1987, Part 1 (Brunswick, ME, 1987). Proceedings of Symposia in Pure Mathematics, vol. 49, pp. 191–237. American Mathematical Society, Providence (1989)Google Scholar
- 27.Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys.
**147**, 1–23 (1992)MathSciNetCrossRefGoogle Scholar - 28.Kontsevich, M., Manin, Y.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys.
**164**, 525–562 (1994)MathSciNetCrossRefGoogle Scholar - 29.Krawitz, M.: FJRW rings and Landau-Ginzburg mirror symmetry, eprint arXiv:0906.0796
- 30.Liu, S.-Q., Wu, C.-Z., Zhang, Y.: On the Drinfeld-Sokolov hierarchies of D type. Int. Math. Res. Notices
**2011**, 1952–1996 (2011)MathSciNetGoogle Scholar - 31.Liu, S.-Q., Yang, D., Zhang, Y.: Uniqueness theorem of \({\cal{W}}\)-constraints for simple singularities. Lett. Math. Phys.
**103**, 1329–1345 (2013)MathSciNetCrossRefGoogle Scholar - 32.Liu, S.-Q., Zhang, Y.: Deformations of semisimple bihamiltonian structures of hydrodynamic type. J. Geom. Phys.
**54**, 427–453 (2005)MathSciNetCrossRefGoogle Scholar - 33.Liu, S.-Q., Zhang, Y.: Jacobi structures of evolutionary partial differential equations. Adv. Math.
**227**, 73–130 (2011)MathSciNetCrossRefGoogle Scholar - 34.Milanov, T., Tseng, H.-H.: Equivariant orbifold structures on the projective line and integrable hierarchies. Adv. Math.
**226**, 641–672 (2011)MathSciNetCrossRefGoogle Scholar - 35.Milanov, T.: Analyticity of the total ancestor potential in singularity theory. Adv. Math.
**255**, 217–241 (2014)MathSciNetCrossRefGoogle Scholar - 36.Okounkov, A., Pandharipande, R.: The equivariant Gromov-Witten theory of \(P^1\). Ann. Math.
**163**, 561–605 (2006)MathSciNetCrossRefGoogle Scholar - 37.Rossi, P.: Gromov-Witten theory of orbicurves, the space of tri-polynomials and symplectic field theory of Seifert fibrations. Math. Ann.
**348**, 265–287 (2010)MathSciNetCrossRefGoogle Scholar - 38.Saito, K.: On a linear structure of the quotient variety by a finite reflexion group. Publ. Res. Inst. Math. Sci.
**29**, 535–579 (1993)MathSciNetCrossRefGoogle Scholar - 39.Saito, K.: Primitive forms for a universal unfolding of a function with an isolated critical point. J. Fac. Sci. Univ. Tokyo Sect. IA Math.
**28**(3) (1981), 775–792 (1982)Google Scholar - 40.Teleman, C.: The structure of 2D semi-simple field theories. Invent. Math.
**188**, 525–588 (2012)MathSciNetCrossRefGoogle Scholar - 41.Witten, E.: Two-dimensional gravity and intersection theory on the moduli space. In: Surveys in Differential Geometry (Cambridge, MA, 1990), pp. 243–310. Lehigh University, Bethlehem (1991)Google Scholar
- 42.Witten, E.: Algebraic geometry associated with matrix models of two-dimensional gravity. In: Topological Models in Modern Mathematics (Stony Brook, NY, 1991) pp. 235–269. Publish or Perish, Houston (1993)Google Scholar
- 43.Witten, E.: Private communicationGoogle Scholar
- 44.Wu, C.-Z.: A remark on Kac-Wakimoto hierarchies of D-type. J. Phys. A
**43**, 035201, 8 pp (2010)Google Scholar - 45.Wu, C.-Z.: Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchie. eprint arXiv:1203.5750
- 46.Zhang, Y.: On the \(CP^1\) topological sigma model and the Toda lattice hierarchy. J. Geom. Phys.
**40**, 215–232 (2002)MathSciNetCrossRefGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2014