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On the Drinfeld-Sokolov hierarchy of type \(E_6^{(1)}\) and its topological solution

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Abstract

In this paper we report some explicit evolutionary PDEs of the Drinfeld-Sokolov hierarchy of type \(E_6^{(1)}\), and show how the unknown functions in these PDEs are related to the tau function. Moreover, for this hierarchy we compute its topological solution of formal series up to a certain degree, whose coefficients of monomials give the Fan-Jarvis-Ruan-Witten invariants for the E6 simple singularity. Based on such results we also derive several explicit evolutionary PDEs and some low-degree terms of the topological solution for the Drinfeld-Sokolov hierarchy of type \(F_4^{(1)}\).

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11601534, 11771461 and 11831017). The authors thank Mattia Cafasso, Jianxun Hu for helpful discussions. The authors also thank the referees for their suggestions to improve the presentation of the manuscript.

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Correspondence to Chao-Zhong Wu.

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He, W., Ke, H. & Wu, C. On the Drinfeld-Sokolov hierarchy of type \(E_6^{(1)}\) and its topological solution. Sci. China Math. (2020). https://doi.org/10.1007/s11425-018-9568-x

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Keywords

  • Drinfeld-Sokolov hierarchy
  • tau function
  • topological solution

MSC(2010)

  • 37K10
  • 17B80