Witten’s top Chern class on the moduli space of higher spin curves

Abstract

This paper is a sequel to [9]. Its goal is to verify that the virtual top Chern class c 1/r in the Chow group of the moduli space of higher spin curves \(\overline M _{{g,n}}^{{1/r}} \) constructed in [9], satisfies all the axioms of spin virtual class formulated in [5]. Hence, according to [5], it gives rise to a cohomological field theory in the sense of Kontsevich-Manin [7]. As was observed in [9], the only non-trivial axioms that have to be checked for the class c 1/r are two axioms that we call Vanishing axiom and Ramond factorization axiom. The first of them requires c 1/r to vanish on all the components of the moduli space \(\overline M _{{g,n}}^{{1/r}} \) where one of the markings is equal to r - 1. The second demands vanishing of the push-forward of c 1/r restricted to the components of the moduli space corresponding to the so called Ramond sector, under some natural finite maps.KeywordsModulus SpaceVector BundleAmple Line BundleChow GroupSpinor BundleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.