Algebraic construction of Witten's top Chern class (Q2752198)

The article deals with the compactified moduli space \(\overline{\mathcal M}_{g,n}^{1/r}\) of stable algebraic curves of genus \(g\) with \(n\) marked points and an \(r\)-spin structure, (i.e. a choice of an \(r\)-th root of the canonical bundle of the curve). Conjecturally [\textit{E. Witten}, Topological methods in modern mathematics, 235--269 (1993; Zbl 0812.14017)], its intersection theory should be related to the Gelfand-Dickey hierarchy. To this end there should exists a certain virtual top Chern class \(c^{1/r}\), with respect to it the intersection numbers are calculated. \textit{T. J. Jarvis}, \textit{T. Kimura} and \textit{A. Vaintrob} [Compos. Math. 126, 157--212 (2001; Zbl 1015.14028)] formulated axioms to be fulfilled by such a class. This class is called spin virtual class. For the case \(g=0\) (and \(r\) arbitrary) and \(r=2\) (and \(g\) arbitrary) they also showed its existence.NEWLINENEWLINEIn the article under review the authors give an algebraic construction of a Chow cohomology class which they propose as candidate for the spin virtual class. Their class fulfills all axioms of the above mentioned article, besides possibly the vanishing axiom. It is shown that for the cases considered by Jarvis, Kimura, and Vaintrob it gives the same answer. A modification of MacPherson's graph construction for the case of 2-periodic (unbounded) complexes is given. It is used to define an analogue of the localized top Chern class for an orthogonal bundle with an isotropic section. This is applied to orthogonal bundles related to families of \(r\)-spin structures, yielding the required virtual class. Some geometric consequences of the vanishing of the virtual top Chern class in the case of 2-spin structures are given.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00024].