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Formal pseudodifferential operators and Witten's r-spin numbers

Kefeng Liu Ravi Vakil Hao Xu

Algebraic Geometry mathscidoc:1912.43095

arXiv preprint arXiv:1112.4601, 2017.12
We derive an effective recursion for Witten's r-spin intersection numbers, using Witten's conjecture relating r-spin numbers to the Gel'fand-Dikii hierarchy (Theorem 4.1). Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions: Propositions 5.2 and 5.4, Corollary 5.5). We use these closed-form descriptions to prove Harer-Zagier's formula for the Euler characteristic of M_ {g, 1}. Finally in Section 6, we extend Witten's series expansion formula for the Landau-Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brezin and Hikami.
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@inproceedings{kefeng2017formal,
  title={Formal pseudodifferential operators and Witten's r-spin numbers},
  author={Kefeng Liu, Ravi Vakil, and Hao Xu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221111516419641655},
  booktitle={arXiv preprint arXiv:1112.4601},
  year={2017},
}
Kefeng Liu, Ravi Vakil, and Hao Xu. Formal pseudodifferential operators and Witten's r-spin numbers. 2017. In arXiv preprint arXiv:1112.4601. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221111516419641655.
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