Formal pseudodifferential operators and Witten's r-spin numbers

2011/12/20 by Kefeng Liu, Ravi Vakil, Liu, Kefeng +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.1112.4601

openalex publication_date 2011/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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