Formal pseudodifferential operators and Witten’s r-spin numbers
| Autor: | Kefeng Liu, Ravi Vakil, Hao Xu |
|---|---|
| Rok vydání: | 2014 |
| Předmět: |
Pure mathematics
Conjecture Applied Mathematics General Mathematics 010102 general mathematics Zero (complex analysis) 01 natural sciences Section (fiber bundle) symbols.namesake Intersection Euler characteristic Genus (mathematics) 0103 physical sciences symbols 010307 mathematical physics 0101 mathematics Gamma function Series expansion Mathematics |
| Zdroj: | Journal für die reine und angewandte Mathematik (Crelles Journal). 2017:1-33 |
| ISSN: | 1435-5345 0075-4102 |
| Popis: | 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. Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions). We use these closed-form descriptions to prove Harer–Zagier’s formula for the Euler characteristic of ℳ g , 1 {\mathcal{M}_{g,1}} . Finally, 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 Brézin and Hikami. |
| Databáze: | OpenAIRE |
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