Formal pseudodifferential operators and Witten's r-spin numbers

TL;DR

This paper develops a recursion for Witten's r-spin intersection numbers using pseudodifferential operators, providing explicit formulas and extending Witten's series expansion to genus zero, thereby advancing the understanding of r-spin theories.

Contribution

It introduces a new recursion for r-spin numbers based on pseudodifferential operators and derives closed-form expressions, extending Witten's series expansion to genus zero.

Findings

Derived an effective recursion for r-spin intersection numbers

Provided closed-form formulas involving gamma functions

Extended Witten's series expansion to genus zero in the small phase space

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.