Borodin–Okounkov formula, string equation and topological solutions of Drinfeld–Sokolov hierarchies

  • Mattia CafassoEmail author
  • Chao-Zhong Wu


We give a general method to compute the expansion of topological tau functions for Drinfeld–Sokolov hierarchies associated with an arbitrary untwisted affine Kac–Moody algebra. Our method consists of two main steps: first, these tau functions are expressed as (formal) Fredholm determinants of the type appearing in the Borodin–Okounkov formula, and then, the kernels for these determinants are found using a reduced form of the string equation. A number of explicit examples are given.


Kac-Moody algebras Integrable systems Drinfeld-Sokolov hierarchies FJRW theory 

Mathematics Subject Classification

37K10 17B67 



The author M. C. thanks the School of Mathematics and Computational Science of the Sun Yat-Sen University for the warm hospitality during the preparation of this article; C.-Z. W. thanks Youjin Zhang, Si-Qi Liu, Di Yang and Jianxun Hu for helpful discussions.

Funding The funding was provided by National Natural Science Foundation of China (Grand Nos. 11771461 and 11831017), Agence Nationale de la Recherche (Grand No. BLAN012004) and European Union Horizon 2020 (Grand No. 778010).


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.LAREMA, UMR 6093, CNRS, SFR Math-STICUNIV AngersAngersFrance
  2. 2.School of MathematicsSun Yat-Sen UniversityGuangzhouPeople’s Republic of China

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