Open \(r\)-spin theory. II: The analogue of Witten's conjecture for \(r\)-spin disks
The paper under review forms part of a series of three articles devoted to the construction of \(r\)-spin theory for Riemann surfaces with boundary. Part I was published in [Int. Math. Res. Not. 2022, No. 14, 10458--10532 (2022; Zbl 1504.14078)], and Part III in [J. Geom. Phys. 192, Article ID 104960, 12 p. (2023; Zbl 1523.14009)].\N\NLet \((C;z_1, \dots , z_n)\) be a smooth marked curve. Then, an \(r\)-spin structure is a line bundle \(S\) together with an isomorphism \(S^{\otimes r} \approx \omega_C(- \sum_{i=1}^n a_i[z_i])\), where \(a_i \in \{0,1, \dots ,r-1\}\). In Part I the authors gave the foundation for an \(r\)-spin theory in genus zero for Riemann surfaces with boundary. The authors restricted themselves in that paper to such surfaces in which each connected component has genus zero.\N\NIn the present paper, they define open \(r\)-spin intersection numbers and prove topological recursion relations that these numbers satisfy. Then these relations are used to prove the main results, Theorems 1.1, 1.2 and 1.4, which provide the open \(r\)-spin version of Witten's conjecture in genus zero. So, this paper concludes the construction for genus zero.\N\NIn Part III, which in fact appeared earlier, the authors study the extension of these results to higher genus.
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